chow rings, decomposition of the diagonal and the topology...

Chow rings, decomposition of the diagonal

and the topology of families

Claire VoisinCNRS and Institut de mathematiques de Jussieu

Contents

0 Introduction 50.1 Decomposition of the diagonal and spread . . . . . . . . . . . . . 7

0.1.1 Spread . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70.1.2 Spreading out rational equivalence . . . . . . . . . . . . . 80.1.3 Applications to Mumford’s type theorems . . . . . . . . . 90.1.4 Another spreading principle . . . . . . . . . . . . . . . . . 10

0.2 Applications to Bloch’s generalized conjecture . . . . . . . . . . . 110.3 Small diagonal and topology of families . . . . . . . . . . . . . . 130.4 Integral coefficients and birational invariants . . . . . . . . . . . 150.5 Organization of the text . . . . . . . . . . . . . . . . . . . . . . . 16

1 Review of Hodge theory and algebraic cycles 191.1 Chow groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.1.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . 191.1.2 Localization exact sequence . . . . . . . . . . . . . . . . . 201.1.3 Functoriality and motives . . . . . . . . . . . . . . . . . . 221.1.4 Cycle class . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.2 Hodge structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.2.1 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . 281.2.2 Hodge classes . . . . . . . . . . . . . . . . . . . . . . . . 301.2.3 Standard conjectures . . . . . . . . . . . . . . . . . . . . . 301.2.4 Mixed Hodge structures . . . . . . . . . . . . . . . . . . . 341.2.5 Coniveau . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2 Decomposition of the diagonal 392.1 A general principle . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.1.1 Mumford’s theorem . . . . . . . . . . . . . . . . . . . . . 422.1.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.2 Varieties with small Chow groups . . . . . . . . . . . . . . . . . . 462.2.1 Generalized decomposition of the diagonal . . . . . . . . . 462.2.2 Generalized Bloch conjecture . . . . . . . . . . . . . . . . 482.2.3 Nilpotence conjecture and Kimura’s theorem . . . . . . . 50

3

4 CONTENTS

3 Complete intersections of large coniveau 553.1 Hodge coniveau of complete intersections . . . . . . . . . . . . . 55

3.1.1 Proof of Theorem 3.1.1 in the case of hypersurfaces . . . . 593.1.2 Complete intersections . . . . . . . . . . . . . . . . . . . . 613.1.3 Coniveau 1 and further examples . . . . . . . . . . . . . 64

3.2 Coniveau 2 complete intersections . . . . . . . . . . . . . . . . . 643.2.1 A conjecture on effective cones . . . . . . . . . . . . . . . 643.2.2 On the generalized Hodge conjecture for coniveau 2 com-

plete intersections . . . . . . . . . . . . . . . . . . . . . . 653.3 On the generalized Bloch and Hodge conjectures . . . . . . . . . 67

3.3.1 Varieties with trivial Chow groups . . . . . . . . . . . . . 683.3.2 A consequence of Conjecture 1.2.10 . . . . . . . . . . . . . 703.3.3 A spreading result . . . . . . . . . . . . . . . . . . . . . . 703.3.4 Proof of theorem 3.3.2 . . . . . . . . . . . . . . . . . . . . 713.3.5 Further applications . . . . . . . . . . . . . . . . . . . . . 77

4 Chow ring of K3 surfaces 854.1 Canonical ring of a K3 surface . . . . . . . . . . . . . . . . . . . 85

4.1.1 Other hyper-Kahler manifolds . . . . . . . . . . . . . . . . 884.2 A decomposition of the small diagonal . . . . . . . . . . . . . . . 92

4.2.1 Calabi-Yau hypersurfaces . . . . . . . . . . . . . . . . . . 974.3 Deligne’s decomposition theorem . . . . . . . . . . . . . . . . . . 101

4.3.1 Deligne’s decomposition theorem . . . . . . . . . . . . . . 1014.3.2 Multiplicative decomposition isomorphisms . . . . . . . . 1044.3.3 Families of abelian varieties . . . . . . . . . . . . . . . . . 1074.3.4 A multiplicative decomposition theorem for families of K3

surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5 Integral coefficients 1195.1 Integral Hodge classes and birational invariants . . . . . . . . . . 119

5.1.1 Atiyah-Hirzebruch-Totaro topological obstruction. . . . . 1195.1.2 Kollar’s example . . . . . . . . . . . . . . . . . . . . . . . 121

5.2 RC varieties and the rationality problem . . . . . . . . . . . . . . 1235.2.1 The group Z2n−2(X) . . . . . . . . . . . . . . . . . . . . 1255.2.2 The group Z4(X) and unramified cohomology . . . . . . 128

5.3 Integral decomposition of the diagonal . . . . . . . . . . . . . . . 1335.3.1 Structure of the Abel-Jacobi map and birational invariants 1335.3.2 Decomposition of the diagonal and structure of the Abel-

Jacobi map . . . . . . . . . . . . . . . . . . . . . . . . . . 140

Chapter 0

Introduction

These lectures are devoted to the interplay between cohomology and Chowgroups, and also to the consequences, on the topology of a family of smoothprojective varieties, of statements concerning Chow groups of the general orvery general fiber.

A crucial notion is that of coniveau of cohomology : A Betti cohomologyclass has geometric coniveau ≥ c if it is supported on a closed algebraic subsetof codimension ≥ c. The coniveau of a class of degree k is ≤ k

2 . As a smoothprojective variety X has non zero cohomology in degrees 0, 2, 4, . . . obtained bytaking c1(L), with L an ample line bundle on X, and its powers c1(L)i, it is notexpected that the whole cohomology of X has large coniveau. But it is quitepossible that the “transcendental” cohomology H∗

B(X)⊥alg consisting of classesorthogonal with respect to the Poincare pairing, to cycle classes on X has largeconiveau.

There is another notion of coniveau, the Hodge coniveau which is com-puted by looking at the shape of the Hodge structures on H∗

B(X,Q). Classesof algebraic cycles are conjecturally detected by Hodge theory as Hodge classes,which are the degree 2k rational cohomology classes of Hodge coniveau k. Thegeneralized Hodge conjecture due to Grothendieck identifies more generally theconiveau above (or geometric coniveau) to the Hodge coniveau.

The next crucial idea goes back to Mumford [61], who observed that for asmooth projective surface S, there is a strong correlation between the structureof the group CH0(S) of 0-cycles on S modulo rational equivalence and thespaces of holomorphic forms on S. The degree 1 holomorphic forms governthe Albanese map and thus allow to detect a certain quotient of the groupCH0(S)hom of 0-cycles homologous to 0, (that is of degree 0 if S is connected,)which is in fact an abelian variety. This part of CH0(S)hom is small in different(but equivalent) senses: First of all because it is parameterized by an algebraicgroup, and secondly because for any ample curve C ⊂ S, the composite map

CH0(C)hom → CH0(S)hom → Alb(S)

is surjective. Thus 0-cycles supported on a given curve suffice to exhaust this

5

6 CHAPTER 0. INTRODUCTION

part of CH0(S)hom.Mumford’s theorem [61] says the following:

Theorem 0.0.1. If H2,0(S) 6= 0, no curve Cj

→ S satisfies the property thatj∗ : CH0(C) → CH0(S) is surjective.

The parallel with geometric coniveau in cohomology is obvious in this case:indeed, the assumption that H2,0(S) 6= 0 is equivalent by Lefschetz theoremon (1, 1)-classes to the fact that the cohomology H2

B(S,Q) is not supportedon a divisor of S. Thus Mumford’s theorem exactly says that if the degree 2cohomology of S is not supported on any divisor, then its Chow group CH0(S)is not supported on any divisor.

The converse to such statement is the famous Bloch conjecture [13]. TheBloch conjecture has been generalized in various forms, one involving filtra-tions on Chow groups, the graded pieces of the filtration being governed bythe coniveau of Hodge structures of adequate degree (cf. [75]). The crucialproperties of this conjectural filtration are functoriality under correspondences,finiteness, and the fact that correspondences homologous to 0 shift the filtration.

We will focus in these notes to a more specific higher dimensional general-ization of the Bloch conjecture, “the generalized Bloch conjecture”, which saysthat if the cohomology H∗

B(X,Q)⊥alg has coniveau ≥ c, then the cycle class mapcl : CHi(X)Q → H2n−2i

B (X,Q) is injective for i ≤ c − 1. In fact, if the varietyX has dimension > 2, there are two versions of this conjecture, according towhether we consider the geometric or the Hodge coniveau. Of course, the twoversions are equivalent assuming the generalized Hodge conjecture. In section3.3 we will prove following [94] this conjecture for the geometric coniveau andfor very general complete intersections of ample hypersurfaces in a smooth pro-jective variety X with “trivial” Chow groups, that is, having the property thatthe cycle class map

cl : CH∗(X)Q → H2∗B (X,Q)

is injective (hence an isomorphism according to [57]).A completely different approach to such statements had been initiated by

Kimura, and it works concretely for those varieties which are dominated byproducts of curves. It should be mentioned here that all we said before works aswell in the case of motives (cf. section 1.1.3). In the above work of Kimura, onecan replace “varieties which are dominated by products of curves” by “motiveswhich are a direct summand of the motive of a product of curves”. In our work[94], we can work with a variety X endowed with the action of a finite group Gand consider the submotives of G-invariant complete intersections obtained byconsidering the projectors associated to characters G → 1,−1.

An important tool introduced by Bloch and Srinivas in [14] and allowing torelate informations concerning Chow groups CHi(X) for small i and the geo-metric coniveau of X is the so-called decomposition of the diagonal, which theyinitially considered in its simplest form starting from information on CH0(X),and which has been subsequently generalized in [67], [56] to a generalized de-composition of the diagonal. This leads to an elegant proof of the so-called

0.1. DECOMPOSITION OF THE DIAGONAL AND SPREAD 7

generalized Mumford-Roitman theorem, stating that if the cycle class map

cl : CHi(X)Q → H2n−2iB (X,Q)

is injective for i ≤ c − 1 then the cohomology H∗B(X,Q)⊥alg has geometric

coniveau ≥ c. (The generalized Bloch conjecture is thus the converse to thisstatement.)

The study of the diagonal will play a crucial role in our proof of the general-ized Bloch conjecture for very general complete intersections. The diagonal willappear in a rather different context in Chapter 4, where we will describe ourjoint work with Beauville and further developments concerning the Chow ringsof K3 surfaces and hyper-Kahler manifolds. Here we will be concerned not withthe diagonal ∆X ⊂ X ×X but with the small diagonal ∆ ∼= X ⊂ X ×X ×X.The reason is the fact that ∆ can be seen as a correspondence from X × Xto X, and that it governs among other things the ring structure of CH∗(X).We obtained in [10] a decomposition of ∆ involving the large diagonals and acanonical 0-cycle o constructed in loc. cit..

We will show in section 4.3 an unexpected consequence, obtained in [95], ofthis study combined with the basic spreading principle described in Section 2.1,concerning the topology of families of K3 surfaces.

In a rather different direction, we present in the last chapter recent resultsconcerning Chow groups and Hodge classes with integral coefficients. Playing onthe defect of the Hodge conjecture for integral Hodge classes (cf. [4]), we exhibita number of birational invariants, vanishing for rational projective varieties andof torsion for unirational varieties. Among them is precisely the failure of theBloch-Srinivas diagonal decomposition with integral coefficients: In general,under the assumption that CH0(X) is small, only a multiple of the diagonal ofX can be decomposed as a cycle in X ×X. The minimal such multiple appearsto be an interesting birational invariant of X.

In the rest of this introduction, we survey a little more precisely the mainideas and results presented in this monograph.

Thanks. I thank the Institute for Advanced Study for inviting me to deliverthe Hermann Weyl lectures and for giving me the opportunity to write-up andpublish these notes. I also thank Lie Fu for his careful reading and corrections.

0.1 Decomposition of the diagonal and spread

0.1.1 Spread

The notion of spread of a cycle is the following (see also [39]) : Assume that wehave a family of smooth algebraic varieties, that is a smooth surjective morphism

π : X → B

with geometric generic fiber Xη and closed fiber Xs. If we have a cycle Z ∈Zk(Xη), then we can find a finite cover U → U of a Zariski open set U of B such


that Z is the restriction to the geometric generic fiber of a cycle ZU ∈ Zk(XU ).If we are over C, we can speak of the spread of a cycle Zs ∈ Zk(Xs), where

s ∈ B is a very general point. Indeed, we may assume that π is projective.We know that there are countably many relative Hilbert schemes Mi → Bparameterizing all subschemes in fibers of π. Cycles Z =

∑i niZi in the fibers

of π are similarly parameterized by countably many varieties πJ : NJ → Bwhere the πJ ’s are proper, and the indices J also encode the multiplicities ni.

Let B′ ⊂ B be the complement of the union ∪J∈EIm πJ , where E is the setof indices J for which πJ is not surjective. A point of B′ is a very general pointof B, and by construction of B′, for any s ∈ B′, and any cycle Zs ∈ Zk(Xs),there exist an index J such that MJ → B is surjective, and a point s′ ∈ MJ

such that πJ(s′) = s and the fiber ZMJ ,s′ at s′ of the universal cycle

ZMJ ⊂ XMJ = X ×B MJ

parameterized by MJ is the cycle Zs. By taking linear sections, we can thenfind M ′

J ⊂ MJ , with s′ ∈ M ′J , such that the morphism M ′

J → B is dominatinggenerically finite. The restriction ZM ′

Jof the universal cycle ZMJ

to X ×B M ′J

is then a spread of Zs.

0.1.2 Spreading out rational equivalence

Let π : X → B be a smooth projective morphism, where B is smooth irreducibleand quasi-projective, and let Z ⊂ X be a codimension k cycle. Let us denoteby Zt ⊂ Xt the restriction of Z to the fiber Xt. We refer to chapter 1 for thebasic notions concerning rational equivalence, Chow groups and cycle classes.

An elementary but fundamental fact is the following result, proved in section2.1 is the following:

Theorem 0.1.1. If for any t ∈ B the cycle Zt is rationally equivalent to 0,there exist a Zariski open set U ⊂ B and a nonzero integer N such that NZ|XU

is rationally equivalent to 0, where XU := π−1(U).

Note that the set of points t ∈ B such that Zt is rationally equivalent to 0is a countable union of closed algebraic subsets of B, so that we could in theabove statement, by a Baire category argument, make the a priori weaker (butin fact equivalent) assumption that Zt is rationally equivalent to 0 for a verygeneral point of B.

This statement is what we call the spreading out phenomenon for rationalequivalence. This phenomenon does not occur for weaker equivalence relationssuch as algebraic equivalence.

An immediate but quite important corollary is the following:

Corollary 0.1.2. In the situation of item (ii) of the above theorem, there existsa dense Zariski open set U ⊂ B such that the Betti cycle class [Z] ∈ H2k

B (X ,Q)vanishes on the open set XU .

0.1. DECOMPOSITION OF THE DIAGONAL AND SPREAD 9

Theorem 0.1.1, (ii) applied to the case where the family X → B is trivialthat is X ∼= X ×B leads to the so-called diagonal decomposition principle dueto Bloch-Srinivas [14].

In this case, the cycle Z ⊂ B × X can be seen as a family of cycles on Xparameterized by B or as a correspondence between B and X. Then Theorem0.1.1, (ii) says that if a correspondence Z ⊂ B ×X induces the trivial map

CH0(B) → CHk(X), b 7→ Zb,

then the cycle Z vanishes up to torsion over some open set of the form U ×X,where U is a dense Zariski open set of B.

The first instance of the diagonal decomposition principle appears in [14].This is the case where X = Y \W , with Y smooth and projective, and W ⊂ Yis a closed algebraic subset, B = Y and Z is the restriction to Y × (Y \W ) ofthe diagonal of Y . In this case, to say that the map

CH0(B) → CH0(X), b 7→ Zb,

is trivial is equivalent to say by the localization exact sequence (1.1.2) that anypoint of Y is rationally equivalent to a 0-cycle supported on W . The conclusionis then the fact that the restriction of the diagonal cycle ∆ to a Zariski open setU×(Y \W ) of Y ×Y is of torsion, for some dense Zariski open set U ⊂ Y . Usingthe localization exact sequence, one concludes that a multiple of the diagonal isrationally equivalent in Y × Y to the sum of a cycle supported on Y ×W anda cycle supported on D× Y , where D := Y \U . Passing to cohomology, we getthe following consequence:

Corollary 0.1.3. If Y is smooth projective of dimension n and CH0(Y ) issupported on W ⊂ Y , the class [∆Y ] ∈ H2n

B (Y × Y,Q) decomposes as

[∆Y ] = [Z1] + [Z2]

where the cycles Zi are cycles with Q-coefficients on Y × Y , Z1 is supported onD × Y for some proper closed algebraic subset D $ Y and Z2 is supported onY ×W .

0.1.3 Applications to Mumford’s type theorems

In the Bloch-Srinivas paper [14], an elegant proof of Mumford’s theorem 0.0.1is provided, together with the following important generalization.

Theorem 0.1.4. Let X be a smooth projective variety and W ⊂ X a closed al-gebraic subset of dimension ≤ k such that any point of X is rationally equivalentto a 0-cycle supported on W . Then H0(X, Ωl

X) = 0 for l > k.

This theorem, together with other very important informations concerningthe coniveau (cf. section 1.2.5) of the cohomology of X, is obtained using onlythe cohomological decomposition of the diagonal of X, that is Corollary 0.1.3.


Theorem 0.1.1 (ii), is also the only ingredient in the proof of the general-ized decomposition of the diagonal (cf. [56], [67], [86, ??] and section 2.2.1).The Bloch-Srinivas decomposition of the diagonal described in the previoussubsection is a decomposition (up to torsion and modulo rational equivalence)involving two pieces, one supported via the first projection over a divisor ofX, the other supported via the second projection over a subset W ⊂ X. It isobtained under the condition that CH0(X) is supported on W . The generalizeddecomposition of the diagonal is the following statement, where X is smoothprojective of dimension n :

Theorem 0.1.5. (Theorem 2.2.1) Assume that for k < c, the cycle class maps

cl : CHk(X)⊗Q→ H2n−2kB (X,Q)

are injective. Then there exists a decomposition

m∆X = Z0 + . . . Zc−1 + Z ′ ∈ CHn(X ×X), (0.1.1)

where m 6= 0 is an integer, Zi is supported in W ′i×Wi with dim Wi = i, dim W ′

i =n− i, and Z ′ is supported in T ×X, where T ⊂ X is a closed algebraic subsetof codimension ≥ c.

We refer to section 2.2.1 for applications of this theorem. In fact, the mainapplication involves only the corresponding cohomological version of the decom-position (0.1.1), that is the generalization of Corollary 0.1.3, which concernedthe case c = 1. It implies that under the same assumptions, the transcendentalcohomology H∗

B(X,Q)⊥alg has geometric coniveau ≥ c.Other applications involve the decomposition (0.1.1) in the group CH(X ×

X)/alg of cycles modulo algebraic equivalence. This is the case of applicationsto the vanishing of positive degree unramified cohomology with Q-coefficients(and in fact with Z-coefficients (cf. [5], [22], [14]) thanks to the Bloch-Katoconjecture proved by Rost and Voevodsky, cf. [82]).

0.1.4 Another spreading principle

Although elementary, the following spreading result proved in section 3.3.3 iscrucial for our proof of the equivalence of the generalized Bloch and Hodgeconjectures for very general complete intersections in varieties with trivial Chowgroups (cf. [94] and section 3.3). Its proof is based as usual on the countabilityof the relative Hilbert schemes for a smooth projective family Y → B.

Let π : X → B be a smooth projective morphism and let (π, π) : X ×B X →B be the fibered self-product of X over B. Let Z ⊂ X ×B X be a codimensionk algebraic cycle. We denote the fibers Xb := π−1(b), Zb := Z|Xb×Xb

.

Theorem 0.1.6. (Voisin [94], see also Proposition 3.3.10.) Assume that fora very general point b ∈ B, there exist a closed algebraic subset Yb ⊂ Xb × Xb

of codimension c, and an algebraic cycle Z ′b ⊂ Yb × Yb with Q-coefficients, suchthat

[Z ′b] = [Zb] in H2kB (Xb ×Xb,Q).

0.2. APPLICATIONS TO BLOCH’S GENERALIZED CONJECTURE 11

Then there exist a closed algebraic subset Y ⊂ X of codimension c, and a codi-mension k algebraic cycle Z ′ with Q-coefficients on X ×B X , such that Z ′ issupported on Y ×B Y, and for any b ∈ B


0.2 Applications to Bloch’s generalized conjec-ture

As we will explain in section 2.2.1, the generalized decomposition of the diagonal(Theorem 0.1.5, or rather its cohomological version) leads to the following result:

Theorem 0.2.1. (cf. Theorem 2.2.3) Let X be a smooth projective variety ofdimension m. Assume that the cycle class map

cl : CHi(X)Q → H2m−2iB (X,Q)

is injective for i ≤ c − 1. Then we have Hp,q(X) = 0 for p 6= q and p < c (orq < c).

The Hodge structures on HkB(X,Q)⊥ alg are thus all of Hodge coniveau ≥ c;

in fact they even are of geometric coniveau ≥ c, that is, these Hodge structuressatisfy the generalized Hodge conjecture 1.2.21 for coniveau c.

The generalized Bloch conjecture is the converse to this statement (it canbe as well generalized to motives). One way to state it is the following:

Conjecture 0.2.2. Assume conversely that Hp,q(X) = 0 for p 6= q and p < c(or q < c). Then the cycle class map


is injective for i ≤ c− 1.

However, as a consequence of Theorem 0.2.1 above, this formulation alsocontains a positive solution to the generalized Hodge conjecture saying underthe above assumptions that the transcendental part of the cohomology of X issupported on a closed algebraic subset of codimension c.

A restricted version of the Bloch conjecture (which is equivalent for surfacesor motives of surfaces) is thus the following :

Conjecture 0.2.3. Let X be a smooth projective complex variety of dimensionm. Assume that the transcendental cohomology H∗

B(X,Q)⊥alg is supported ona closed algebraic subset of codimension c. Then the cycle class map




Note that Conjecture 0.2.4 below is also an important generalization ofBloch’s conjecture, concerning only 0-cycles and starting with rather differ-ent cohomological assumptions. There is no direct relation between Conjec-tures 0.2.2 and 0.2.4 except for the fact that they coincide for varieties X withHi,0(X) = 0 for all i > 0. Of course they both fit within the general Bloch-Beilinson conjecture on filtrations on Chow groups, and can be considered asconcrete consequences of them.

Conjecture 0.2.4. Let X be a smooth projective variety such that Hi,0(X) = 0

for i > r. Then there exists a dimension r closed algebraic subset Zj

→ X suchthat j∗ : CH0(Z) → CH0(X) is surjective.

Our main result presented in section 3.3 is obtained as a consequence of thespreading principle 0.1.6. It proves conjecture 0.2.3 for very general complete in-tersections in ambient varieties with “trivial Chow groups”, assuming Lefschetzstandard conjecture. The situation is the following: X is a smooth projectivevariety of dimension n which satisfies the property that the cycle map

cl : CHi(X)Q → H2n−2iB (X,Q)

is injective for all i. Let Li, i = 1, . . . , r be very ample line bundles on X.We consider smooth complete intersections Xt ⊂ X of hypersurfaces Xi ∈ |Li|.They are parameterized by a quasiprojective base B.

Theorem 0.2.5. (Voisin 2011, [94] Assume that for the very general pointt ∈ B, the vanishing cohomology Hn−r

B (Xt,Q)van is supported on a codimensionc closed algebraic subset of Xt. Assume also the Lefschetz standard conjecture(in fact Conjecture 1.2.10 would suffice). Then the cycle class map

cl : CHi(Xt)Q → H2n−2r−2iB (Xt,Q)


In applications, this theorem is particularly interesting in the case where Xis the projective space Pn. In this case, the hypersurfaces Xi are characterizedby their degrees di and we may assume d1 ≤ . . . ≤ dr. When n is large comparedto the di’s, the coniveau of Xt is also large due to the following result essentiallydue to Griffiths (cf. [40]), the proof of which will be sketched in section 3.1.

Theorem 0.2.6. A smooth complete intersection Xt ⊂ Pn of r hypersurfacesof degrees d1 ≤ . . . ≤ dr has coniveau ≥ c if and only if

n ≥r∑

i

di + (c− 1)dr.

If we consider motives (cf. section 1.1.3) and in particular those which areobtained starting from a variety X with an action by finite group G, and lookingat invariant complete intersections Xt ⊂ X and motives associated to characters

0.3. SMALL DIAGONAL AND TOPOLOGY OF FAMILIES 13

of G, we get many new examples where the adequate variant of Theorem 0.2.5holds, because a submotive often has a larger coniveau. A typical example isthe case of Godeaux quintic surfaces, which are free quotients of quintic surfacesin P3 invariant under a certain action of G ∼= Z/5Z. The G-invariant part ofH2,0(S) is zero although the quotient surface S/G is of general type, and theBloch conjecture was already proved for the quotient surfaces S/G in [83]. Theproof we give here is much simpler and has a much wider range of applications,but in higher dimension it is conditional to the standard conjecture. In thesurface case (n−r = 2), the standard conjecture will be automatically satisfied.

In section 2.2.3, we will also describe the ideas of Kimura, which lead to re-sults of a similar shape, namely the implication from the generalized Hodge con-jecture (geometric coniveau=Hodge coniveau) to the generalized Bloch conjec-ture (the Chow groups CHi are “trivial” for i smaller than the Hodge coniveau),but for a completely different class of varieties. More precisely, Kimura’s methodapplies to all motives which are direct summands in the motive of a product ofcurves.

0.3 Decomposition of the small diagonal and ap-plication to the topology of families

In Chapter 4, we will exploit the spreading principle 0.1.1 (ii), or rather its coho-mological version, to exhibit a rather special phenomenon satisfied by families ofabelian varieties, K3 surfaces, and conjecturally also by families of Calabi-Yauhypersurfaces in projective space. Let π : X → B be a smooth projective mor-phism. The decomposition theorem, proved by Deligne in [28] as a consequenceof the hard Lefschetz theorem, is the following statement:

Theorem 0.3.1. (Deligne 1968) In the derived category of sheaves of Q-vectorspaces on B, there is a decomposition

Rπ∗Q = ⊕iRiπ∗Q[−i]. (0.3.2)

The question we consider in chapter 4, following [95], is the following:

Question 0.3.2. Given a family of smooth projective varieties π : X → B, doesthere exist a decomposition as above which is multiplicative, that is compatiblewith the morphism

µ : Rπ∗Q⊗Rπ∗Q→ Rπ∗Q

given by cup-product?

As we will see quickly, the answer is generally negative, even for projectivebundle P(E) → B. However, it is affirmative for families of abelian varieties (cf.section 4.3) due to the group structure of the fibers.

The right formulation in order to get a larger range of families satisfyingsuch a property is to ask whether there exists a decomposition isomorphism


which is multiplicative over a Zariski dense open set U of the base B (or moreoptimistically, which is multiplicative locally on B for the Zariski topology).

One of our main results in this chapter is the following (cf. [95], section 4.3):

Theorem 0.3.3. i) For any smooth projective family π : X → B of K3 sur-faces, there exist a nonempty Zariski open subset B0 of B and a multiplicativedecomposition isomorphism as in 4.3.26 for the restricted family π : X 0 → B0.

ii) The class of the diagonal [∆X 0/B0 ] ∈ H4B(X×BX ,Q) belongs to the direct

summand H0(B0, R4(π, π)∗Q) of H4(X 0 ×B0 X 0,Q), for the induced decompo-sition of R(π, π)∗Q.

iii) For any line bundle L on X , there is a Zariski dense open set B0 ⊂ Bsuch that its topological first Chern class ctop

1 (L) ∈ H2B(X ,Q) restricted to X 0

belongs to the direct summand H0(B0, R2π∗Q).

In the second statement, (π, π) : X 0×B0 X 0 → B0 denotes the natural map.A decomposition Rπ∗Q ∼= ⊕iR

iπ∗Q[−i] induces a decomposition

R(π, π)∗Q = ⊕iRi(π, π)∗Q[−i]

by the relative Kunneth isomorphism

R(π, π)∗Q ∼= Rπ∗Q⊗Rπ∗Q.

In statements ii) and iii), we use the fact that a decomposition isomorphism asin (0.3.2) for π : X → B induces a decomposition

Hk(X ,Q) ∼= ⊕p+q=kHp(B, Rqπ∗Q)

(which is compatible with cup-product if the given decomposition isomorphismis).

This result is in fact a formal consequence of the spreading principle (The-orem 0.1.1) and of the following result due to Beauville and the author:

Theorem 0.3.4. (Beauville-Voisin [10]) Let S be a smooth projective K3 sur-face. There exists a 0-cycle o ∈ CH0(S) such that we have the following equality

∆ = ∆12 · o3 + (perm.)− (o1 · o2 · S + (perm.)) in CH4(S × S × S)Q (0.3.3)

Here ∆ ⊂ S×S×S is the small diagonal (x, x, x), x ∈ S and the ∆ij ’s arethe diagonals xi = xj . The class o ∈ CH0(S) is the class of any point belongingto a rational curve in S and the oi’s are its pull-back in CH2(S × S × S)Qvia the various projections. The cycle ∆12 · o3 is thus the algebraic subset(x, x, o), x ∈ S of S × S × S. The term +(perm.) means that we sum overthe permutations of 1, 2, 3.

We will also prove in Section 4.2.1 a partial generalization of this result forCalabi-Yau hypersurfaces.

Our topological application (Theorem 0.3.3 above) involves the structure ofthe cup-product map

Rπ∗Q⊗Rπ∗Q→ Rπ∗Q,

0.4. INTEGRAL COEFFICIENTS AND BIRATIONAL INVARIANTS 15

which is not surprising since the small diagonal, seen as a correspondence fromS×S to itself, governs the cup-product map on cohomology, as well as the inter-section product on Chow groups. For the same reason, the decomposition (0.3.3)is related to properties of the Chow ring of K3 surfaces. These applications aredescribed in Section 4.1. Our partial decomposition result concerning the smalldiagonal of Calabi-Yau hypersurfaces (Theorem 4.2.6) allows in a similar wayto prove the following:

Theorem 0.3.5. (Voisin 2011 [95]) Let X ⊂ Pn be a Calabi-Yau hypersurfacein projective space. Let Zi, Z ′i be cycles on X such that codim Zi + codim Z ′i =n− 1. Then if we have a cohomological relation

∑

i

ni[Zi] ∪ [Z ′i] = 0 in H2n−2B (X,Q)

this relation already holds at the level of Chow groups:∑

i

niZi · Z ′i = 0 in CHn−1(X)Q.

0.4 Integral coefficients and birational invariants

Everything that has been said before was with rational coefficients. This con-cerns Chow groups and cohomology. In fact it is known that the Hodge conjec-ture is wrong with integral coefficients and also that the diagonal decompositionprinciple (Corollary 2.1.9) is wrong with integral coefficients (that is, with aninteger N set equal to 1). But it turns out that some birational invariants canbe constructed out of this. For example, assume for simplicity that CH0(X)is trivial. Then the minimum integer N such that there is a (either Chow the-oretic or cohomological) decomposition of the diagonal in CHn(X ×X), resp.H2n(X ×X,Z), n = dim X

N∆X = Z1 + Z2, resp. N [∆X ] = [Z1] + [Z2], (0.4.4)

where Z1, Z2 are codimension n cycles in X ×X with

SuppZ1 ⊂ T ×X, T X, Z2 = X × x

for some x ∈ X are birational invariants of X. Even the second one, on whichwe will focus, is in general non trivial because, as we will show, it annihilates thetorsion in H3

B(X,Z), and at when dimX ≤ 3, the torsion in the whole integralcohomology H∗

B(X,Z).The groups Z2i(X) defined as

Z2i(X) := Hdg2i(X,Z)/H2iB (X,Z)alg

themselves are birationally invariant for i = 2, i = n−1, n = dim X, as remarkedin [78], [89]. One part of Chapter 5 is devoted to describing recent results


concerning the groups Z2i(X) for X a rationally connected smooth projectivecomplex variety. On one hand, it is proved in [22] that for such an X, thegroup Z4(X) is equal to the third unramified cohomology group H3

nr(X,Q/Z)with torsion coefficients. This result also proved in [5] is a consequence of theBloch-Kato conjecture now proved by Voevodsky and Rost. We will sketch insection 5.2.2 the basic facts from Bloch-Ogus theory (cf. [12]) which are neededto establish this result.

It follows then from the work of Colliot-Thelene and Ojanguren [20] thatthere are rationally connected 6-folds X for which Z4(X) 6= 0. Such examplesare not known in smaller dimension, and we proved that they do not exist indimension 3. We even have the following results:

Theorem 0.4.1. (Voisin 2006, [88]) Let X be a smooth projective 3-fold whichis either uniruled or Calabi-Yau (i.e. with trivial canonical bundle and H1(X,OX) =0). Then Z4(X) = 0.

The Calabi-Yau case has been extended and used in [44] to show the follow-ing:

Theorem 0.4.2. (Horing-Voisin 2010) Let X be a Fano fourfold or a Fanofivefold of index 2. Then the group Z2n−2(X) is trivial, n = dim X. Hence thecohomology H2n−2

B (X,Z) is generated over Z by classes of curves.

It is in fact very likely that the group Z2n−2(X) is trivial for rationallyconnected varieties, as follows from its invariance under deformations and alsospecialization to characteristic p (cf. Theorem 5.2.7 and [97]).

The final section is devoted to the study of the existence of an integral coho-mological decomposition of the diagonal, particularly for a rationally connected3-fold. This is in fact closely related in this case to the integral Hodge con-jecture, in the following way: If X is a rationally connected 3-fold, (or moregenerally any smooth projective 3-fold with h3,0(X) = 0), the intermediate Ja-cobian J(X) is an abelian variety. There is a degree 4 integral Hodge classα ∈ Hdg4(X × J(X),Z) built using the canonical isomorphism

H1(J(X),Z) ∼= H3B(X,Z)/Torsion.

As we will show in Theorem 5.3.16, if X admits an integral cohomologicaldecomposition of the diagonal, that is a decomposition as in (0.4.4) with N = 1,the above class α is algebraic, or equivalently, there is an universal codimension 2cycle on X parameterized by J(X). The existence of such universal codimension2 cycle is not known in general even for rationally connected 3-folds.

0.5 Organization of the text

Chapter 1 is introductory. We will review Chow groups, correspondences andtheir cohomological and Hodge theoretic counterpart. The emphasis will be puton the notion of coniveau and the generalized Hodge conjecture which statesthe equality of geometric and Hodge coniveau.

0.5. ORGANIZATION OF THE TEXT 17

Chapter 2 is devoted to a description of various forms of the “decompositionof the diagonal” and applications of it. This mainly leads to one implicationwhich is well understood now, namely the fact that trivial Chow groups ofdimension ≤ c − 1 implies (geometric) coniveau ≥ c. We state the converseconjecture (generalized Bloch conjecture).

In Chapter 3, we will first describe how to compute the Hodge coniveau ofcomplete intersections. We will then explain a strategy to attack the generalizedHodge conjecture for complete intersections of coniveau 2. The guiding ideathere is that although the powerful method of the decomposition of the diagonalsuggests that computing Chow groups of small dimension is the right way tosolve the generalized Hodge conjecture, it might be better to invert the logicand try to compute the geometric coniveau directly. And indeed, this chapterculminates with the proof of the fact that for very general complete intersections,assuming Conjecture 1.2.10 or Lefschetz standard conjecture, the generalizedHodge conjecture implies the generalized Bloch conjecture: geometric coniveau≥ c implies the triviality of Chow groups of dimension ≤ c− 1.

In the fourth chapter, we turn to the study of the Chow rings of K3 sur-faces and other K-trivial varieties. This study is related to a decomposition ofthe small diagonal of the triple self-product. We finally show consequences ofthis study for the topology of certain K-trivial varieties: K3 surfaces, abelianvarieties and Calabi-Yau hypersurfaces.

The last chapter is devoted in part to the study of the groups Z2i(X) mea-suring the failure of the Hodge conjecture with integral coefficients. Some van-ishing and nonvanishing results are presented, together with a comparison ofthe group Z4(X) with the so-called unramified cohomology of X with torsioncoefficients. We also consider various forms of the existence of an integral coho-mological decomposition of the diagonal (cf. (0.1.1)) of a 3-fold X with trivialCH0-group. We show that an affirmative answer to this question is equivalentto the vanishing of numerous birational invariants of X.

Convention. We work over C and most of the time we will write X forX(C). A general point of X is a complex point x ∈ U(C), where U ⊂ X is aZariski dense open set. When we say that a property is satisfied by a generalpoint, we thus mean that there exists a Zariski open set U ⊂ X such that theproperty is satisfied for any point of U(C). This is not always equivalent tobeing satisfied at the geometric generic point (assuming X is connected), sincesome properties are not Zariski open: a typical example is the following: Letφ : Y → X be a smooth projective morphism with fiber Yt, t ∈ X. Let L be aline bundle on Y . The property that Pic Yt = ZL|Yt

is not Zariski open.A very general point of X is a complex point x ∈ X ′ ⊂ X(C), where X ′ ⊂

X(C) is the complement of a countable union of proper closed algebraic subsetsof X. When we say that a property is satisfied by a very general point, we thusmean that there exists a countable union of dense Zariski open sets Ui suchthat the property is satisfied for any element of ∩iUi(C). For example, in the


situation above, the property that Pic Yt = ZL|Ytis satisfied at a very general

point of X if it is satisfied at the geometric generic point of X.

Chapter 1

Review of Hodge theoryand algebraic cycles

This chapter is introductory. We review the necessary background material usedin the rest of the notes. This includes Chow groups, correspondences and mo-tives on the purely algebraic side, cycle classes, (mixed) Hodge structures on thealgebraic-topological side. The emphasis is put on the notion of Hodge versusgeometric coniveau.

1.1 Chow groups

1.1.1 Construction

We follow [35], [86, II]. Let X be a scheme over a field K, (which in practicewill always be a quasi-projective scheme, i.e. a Zariski open set in a projectivescheme). Let Zk(X) be the group of k-dimensional algebraic cycles of X, i.e.the free abelian group generated by the (reduced and irreducible) closed k-dimensional subvarieties of X defined over K. If Y ⊂ X is a subscheme ofdimension ≤ k, we can associate a cycle c(Y ) ∈ Zk(X) to it as follows: set

c(Y ) =∑

W

nW W, (1.1.1)

where the sum is taken over the k-dimensional irreducible reduced componentsof Y , and the multiplicity nW is equal to the length l(OY,W ) of the Artinianring OY,W , the localisation of OY at the point W .

If φ : Y → X is a proper morphism between quasi-projective schemes, wecan define

φ∗ : Zk(Y ) → Zk(X)

by associating to a reduced irreducible subscheme Z ⊂ Y the cycle deg [K(Z) :K(Z ′)]Z ′, Z ′ = φ(Z), if φ : Z → Z ′ is generically finite, and 0 otherwise (in the

19

20CHAPTER 1. REVIEW OF HODGE THEORY AND ALGEBRAIC CYCLES

latter case we have dim Z ′ < k). The properness is already used here in orderto guarantee that Z ′ is closed in X.

If W is a normal algebraic variety, the localized rings at the points of codi-mension 1 of W are discrete valuation rings, so we can define the divisor div(φ)of a non-zero rational function φ ∈ K(W )∗. This divisor div(φ) is the w − 1-cycle of W , w := dim W defined as

∑codim T=1 νT (φ)T . If W ⊂ X is a closed

subvariety and τ : W → X is the normalisation of W , we have the mapτ∗ : Zk(W ) → Zk(X). This allows us to give the following definition.

Definition 1.1.1. The subgroup Zk(X)rat of cycles rationally equivalent to 0is the subgroup of Zk(X) generated by cycles of the form

τ∗div(φ), dim W = k, φ ∈ K(W )∗,

where τ : W → W ⊂ X is the normalisation of a closed (k + 1)-dimensionalsubvariety of X.

We write CHk(X) = Zk(X)/Zk(X)rat.If X is n-dimensional and smooth, or more generally, locally factorial, then

Zn−1(X) is the group of Cartier divisors, and the group CHn−1(X) can beidentified with the group Pic(X) of algebraic line bundles on X modulo iso-morphism. Indeed, this isomorphism makes the line bundle OX(D) = ⊗iI⊗−ni

Di

correspond to a divisor D =∑

i niDi, and D is rationally equivalent to D′ if andonly if there exists φ ∈ K(X) such that div(φ) = D−D′. Clearly, multiplicationby φ then induces an isomorphism

φ : OX(D) ∼= OX(D′).

Conversely, every algebraic line bundle L admits a meromorphic section, bytrivialization on a Zariski dense open set, and is thus isomorphic to a sheaf ofthe form OX(D).

In general, for X reduced and irreducible of dimension n, we have a map

s : Pic X → CHn−1(X)

which to L associates the class of τ∗div σ, where τ : X → X is the normalisationand σ is a non-zero meromorphic section of τ∗L.

1.1.2 Localization exact sequence

Let X be a quasi-projective scheme, and let Fl→ X be the inclusion of a closed

subscheme. Let j : U = X − F → X be the inclusion of the complement.The morphism l is proper since it is a closed immersion. The morphism j is anopen immersion, and j∗ will be defined by restricting cycles to the open set U .Thus, we have the inverse image morphism j∗ and the direct image morphisml∗. Moreover, it is clear that j∗ l∗ = 0, since the cycles supported on F do notintersect U .

1.1. CHOW GROUPS 21

Lemma 1.1.2. The following sequence, known as the localization sequence, isexact:

CHk(F ) l∗→ CHk(X)j∗→ CHk(U) → 0. (1.1.2)

Proof. The surjectivity on the right follows from the fact that if Z ⊂ U isa k-dimensional subvariety, then its Zariski closure Z ⊂ X is a k-dimensionalsubvariety whose intersection with U is equal to Z.

Furthermore, let Z ∈ Zk(X) be such that j∗Z ∈ Zk(U)rat. Then thereexists Wi ⊂ U, dim Wi = k + 1, φi ∈ K(Wi)∗, and integers ni, such that

Z ∩ U =∑

i

ni(τi)∗(div(φi)), (1.1.3)

where τi : Wi → U is the normalisation of Wi. Then, if Wi is the closure ofWi in X and τ i : Wi → X is its normalisation, we have φi ∈ K(Wi)∗, and theequality (1.1.3) shows that

Z −∑

i

ni(τ i)∗(div(φi)) ∈ Zk(F ).

This proves the exactness of the sequence (1.1.2) in the middle, since Z −∑i ni(τ i)∗(div(φi)) is rationally equivalent to Z in X.

In [35], Fulton proposes an intersection theory

CHk(X)× CHl(X) → CHk+l−n(X)

for a smooth n-dimensional variety X, which is particularly well adapted tocomputing the excess intersection formulae. If Z and Z ′ are two irreduciblereduced subschemes of X, of dimensions k and l respectively, which intersectproperly and generically transversally, i.e. such that dimZ ∩Z ′ = k+ l−n, andgenerically along the intersection Z ∩Z ′, Z and Z ′ are smooth and the intersec-tion is transverse, then one classically defines Z ·Z ′ to be the cycle associated tothe scheme Z ∩Z ′ (which has in fact all its components of multiplicity 1 by as-sumption). By bilinearity, we can define the intersection Z ·Z ′ this way for anypair of cycles whose supports intersect properly and generically transversally.

If Z and Z ′ do not intersect properly, the classical theory replaces Z by acycle Z which is rationally equivalent to Z and intersects Z ′ properly (such acycle exists by “Chow’s moving lemma”), and defines Z · Z ′ to be the class ofZ ∩ Z ′ in CHk+l−n(X).

Let | Z |= ∪iZi denote the support of the cycle Z =∑

i niZi. Fulton’s theoryavoids the recourse to “Chow’s moving lemma”, and gives a refined intersection,i.e. a cycle

Z · Z ′ ∈ CHk+l−n(| Z | ∩ | Z ′ |)for every pair of cycles Z and Z ′, whose image in CHk+l−n(X) is Z · Z ′. Thusit provides an exact answer to the problems of excess, i.e. the “explicit” com-putation of Z · Z ′ as a cycle supported on Sup Z ∩ Sup Z ′ when Z and Z ′ donot intersect properly.


1.1.3 Functoriality and motives

If p : Y → X is a proper morphism of quasi-projective schemes, we have themorphism

p∗ : Zk(Y ) → Zk(X)

defined above.

Lemma 1.1.3. p∗ sends Zk(Y )rat to Zk(X)rat, and thus induces a morphism

p∗ : CHk(Y ) → CHk(X).

Proof. Let τ : W → W ⊂ Y be the normalisation of a closed (k + 1)-dimensional subvariety of Y , and let φ ∈ K(W )∗. Assume first that the com-position p τ : W → X is generically finite, and let W ′ = Im p τ ⊂ X,τ ′ : W ′ → X be its normalisation. We have a factorisation

τ : W//

p

²²

Y

p

²²τ ′ : W ′ // X,

so that p∗ τ∗ = τ ′∗ p∗ : Zk(W ) → Zk(X).The function field K(W ) is an algebraic extension of K(W ′). Consider the

norm morphismN : K(W )∗ → K(W ′)∗

relative to this field extension.

Lemma 1.1.4. For every non-zero rational function on W , we have the equality

p∗(div φ) = div(N(φ)).

This formula and the definition of rational equivalence imply Lemma 1.1.3.

Now assume that p : Y → X is a flat morphism of relative dimensionl = dim Y −dim X. If Z ⊂ X is a reduced irreducible k-dimensional subscheme,then p−1(Z) is a (k + 1)-dimensional subscheme of Y , and thus it admits anassociated cycle p∗Z ∈ Zk+l(Y ). Extending this definition by Z-linearity, wethus obtain p∗ : Zk(X) → Zk+l(Y ).

Lemma 1.1.5. The map p∗ defined above sends Zk(X)rat to Zk+l(Y )rat. Thus,it induces a morphism

p∗ : CHk(X) → CHk+l(Y ).

Remark 1.1.6. Let CHk(X) := CHn−k(X) where X is irreducible of dimen-sion n. Then flat pull-back sends CHk(X) to CHk(Y ).

1.1. CHOW GROUPS 23

Lemma 1.1.5 allows to define the pull-back morphism p∗ when p is flat,which is much too restrictive. A crucial point now is the existence of a pull-back morphism

p∗ : CHk(X) → CHk(Y )

once X is smooth. One uses for this the fact that p factors as pr2 jp, wherepr2 : Y ×X → X is the second projection (which is flat), and jp is the inclusiony 7→ (y, p(y)) of Y into Y ×X. By smoothness of X, the image of jp is a localcomplete intersection in X × Y , and restriction to local complete intersectionsubschemes is well-defined inductively starting from the divisor case (see [35,6.6]). The pull-back map p∗ is then defined as the composition of pr∗2 followedby the restriction map to Im jp = Y .

We have the following compatibility between the intersection product andthe morphisms p∗, p∗, where X, Y are smooth and p : Y → X is a morphism.

Proposition 1.1.7. 1. (projection formula, [35], 8.1) If p is proper, thenfor Z ∈ CH(Y ) and Z ′ ∈ CH(X), we have

p∗(p∗Z · Z ′) = Z · p∗Z ′ ∈ CH(X). (1.1.4)

2. With no hypothesis of properness on p, for Z, Z ′ ∈ CH(X), we have

p∗(Z · Z ′) = p∗Z · p∗Z ′ inCH(Y ).

The following corollary is a consequence of the projection formula.

Corollary 1.1.8. If p : Y → X is a proper morphism with dim X = dim Y ,then

p∗ p∗Z = deg p Z

for Z ∈ CH(X), where deg p is defined to be equal to 0 if p is not dominant.

To prove this, we apply formula (1.1.4) with Z ′ = c(Y ) ∈ CHn(Y ), n =dim Y . Indeed, by the definition of p∗, we have p∗(c(Y )) = deg p c(X) ∈ CH(X).

Definition 1.1.9. A correspondence between two smooth varieties X and Y isa cycle Γ ∈ CH(X × Y ). A 0-correspondence between X and Y is an elementof CHn(X × Y ), n = dim X.

If X is projective, the second projection X×Y → Y is proper. A correspon-dence Γ ∈ CHk(X × Y ) then defines a morphism

Γ∗ : CHi(X) → CHi+k−dim X(Y )

given byΓ∗(Z) = pr2∗(pr∗1(Z) · Γ), Z ∈ CH(X).

Note that if Γ is a 0-correspondence, Γ∗ : CH∗(X) → CH∗(Y ) preserves thedegree.


If Y is projective, we can also consider the morphism

Γ∗ : CH(Y ) → CH(X)

given byΓ∗(Z) = pr1∗(pr∗2(Z) · Γ), Z ∈ CH(Y ).

Now, assume that X, Y and W are smooth varieties, with X and Y projective,and let Γ ∈ CHk(X × Y ), Γ′ ∈ CHk′(Y ×W ) be correspondences. We can thendefine the composition Γ′ Γ ∈ CHk′′(X ×W ), where k′′ = k + k′ − dim Y , bythe formula

Γ′ Γ = p13∗(p∗12Γ · p∗23Γ′), (1.1.5)

where the pij , i = 1, 2, 3 are the projections of X × Y ×W onto the product ofits i-th and j-th factors.

We can show that the composition of correspondences is associative. In par-ticular, it equips the group CH(X×X) with the structure of a (non-commutative)ring.

Finally, we have the following essential formula, which is a consequence ofthe projection formula (1.1.4) (cf. [86, ??]).

Proposition 1.1.10. Let

Γ∗ : CH(X) → CH(Y ), Γ′∗ : CH(Y ) → CH(W )

be the morphisms associated to the correspondences Γ, Γ′. Then

(Γ′ Γ)∗ = Γ′∗ Γ∗.

Motives

The fact that correspondences can be composed suggests to consider a categoryof smooth projective varieties, the morphisms being correspondences moduloa given equivalence relation. A better category is proposed by Grothendieck(cf. [1], [62]). First of all, consider the category of effective motives: Insteadof considering as objects the smooth projective varieties X, one considers thepairs (X, p), where X is a smooth projective variety of dimension n and p ∈CHn(X ×X)Q is a projector, namely

p p = p in CHn(X ×X)Q.

The morphisms between (X, p) and (Y, q) are the 0-correspondences between Xand Y of the form

q γ p, γ ∈ CHn(X × Y ), n = dim X.

Next, general Chow motives are defined as follows: Objects are triple (X, p,m),where X is smooth projective, p ∈ CHdim X(X×X)Q is a projector, and m is an

1.1. CHOW GROUPS 25

integer. Then the morphisms between (X, p,m) and (Y, q, m′) are the elementsof CHdim X+m′−m(X × Y )Q of the form

q γ p, γ ∈ CHdim X+m′−m(X × Y ).

Of course, one can make the theory for other equivalence relations, for ex-ample homological equivalence, and even by introducing more morphisms as in[2], where Andre allows further morphisms by inverting certain Lefschetz opera-tors. (If the Hodge conjecture or even the standard conjectures hold, this is thesame set of morphisms, but his theory works unconditionally and the resultingcategory has better semisimplicity properties, deduced from the correspond-ing semisimplicity properties of the category of polarized Hodge structures (cf.Theorem 1.2.3.)

Example 1.1.11. (Lefschetz motive) Let 0 be any point of P1. One considersthe pair (P1, p) where p is the projector given by P1×0. The independence of thechoice of point is due to the fact that all points on P1 are rationally equivalenton any field, even non algebraically closed.

Note that we can consider sums and tensor products of motives. Sums areinduced (at least for the effective motives (X, p, 0)) by disjoint unions, whiletensor products are induced by usual products. We can thus speak of the self-product of a motive (X, p): This is the motive (X ×X, p× p).

Example 1.1.12. (Powers of the Lefschetz motive) The motive L⊗n is ((P1)n, pn)where pn is the projector given by (P1)n × (0, . . . , 0), where 0 is any point ofP1. As (P1)n is birationally equivalent to Pn, L⊗n is isomorphic to the motive(Pn, pn) where pn is the class of Pn × pt for any point of Pn.

Let us give a few examples:

Example 1.1.13. (Generically finite morphisms) Let X, Y be smooth varietiesof dimension n and

φ : X → Y

be a generically finite morphism. Consider the correspondence ΓY = (φ,φ)∗∆Y

deg φ ∈CHn(X ×X)Q where ∆Y is the diagonal of Y .

Lemma 1.1.14. The correspondence ΓY is a projector, and (X, ΓY ) is isomor-phic to Y .

Proof. We apply the definition of the composition of correspondences, theprojection formula and the fact that φ∗∆X = deg φ∆Y . The isomorphismbetween Y and (X, ΓY ) is given by the transpose of the graph of φ.

We thus get another motive (X, ∆X − ΓY ) which is the motive of X withthe motive of Y removed.

In the above example, (X, ∆X) is the full motive of X, with projector pinduced by the diagonal of X, which acts as the identity on CH(X).


Example 1.1.15. (Finite group actions) Let X be a smooth projective varietyof dimension n and G a finite group acting on X. For any g ∈ G we havethe graph of g, which provides a correspondence Γg ∈ CHn(X ×X). Let nowχ : G → 1,−1 be a character, and consider the following correspondence:

pχ =1|G|

∑

g∈G

χ(g)Γg ∈ CHn(X ×X)Q.

pχ is a projector, due to the fact that Γg Γg′ = Γgg′ .

1.1.4 Cycle class

Assume that X is a smooth complex quasi-projective variety over C. Thus Xcan also be seen as a smooth complex manifold, usually denoted by Xan toemphasize the use of the holomorphic structural sheaf, or Xcl to emphasize theuse of the classical topology. Given a subvariety Z in X, one defines the cycleclass [Z] ∈ H2n−2k

B (X,Z) in the Betti cohomology groups of X (that is thecohomology of X(C) endowed with the classical topology). Let Z be a reducedirreducible subvariety of codimension k in X. By Hironaka’s theorem, there isa desingularization

i : Z −→ Z ⊂ X

of Z, and we may consider

H2n−2k,B(Z,Z) ∼= Z i∗→ H2n−2k,B(X,Z) ∼= H2kB (X,Z),

where the last isomorphism is given by Poincare duality, and the first isomor-phism comes from the fact that Z(C) is a connected compact complex manifold,hence canonically oriented. The image of 1 gives a class

[Z] ∈ H2kB (X,Z).

This is the integral Betti cycle class of Z. In many places we will use therational cycle class [Z] ∈ H2k

B (X,Q).We extend by linearity the above cycle class to any cycle Z =

∑i Zi ∈

Zk(X).

Lemma 1.1.16. If Z is rationally equivalent to 0, then [Z] = 0 in H2kB (X,Z).

The map Z 7→ [Z] thus gives the “cycle class” map

cl : CHl(X) → H2n−2lB (X,Z).

The cycle class map is compatible with the intersection product

· : CHl(X)× CHk(X) → CHk+l(X),

and the cup-product

∪ : H2lB (X,Z)×H2k

B (X,Z) → H2k+2lB (X,Z).

1.2. HODGE STRUCTURES 27

Proposition 1.1.17. For Z ∈ CHl(X), Z ′ ∈ CHk(X), we have

cl(Z · Z ′) = cl(Z) ∪ cl(Z ′) ∈ H2k+2lB (X,Z).

The cycle class map possesses the following functoriality properties.

Proposition 1.1.18. Let i : Y → X be a morphism between smooth varieties.

1. If Z ∈ CHk(X), then

i∗cl(Z) = cl(i∗Z) ∈ H2kB (Y,Z).

2. If i is proper and Z ∈ CHk(Y ), then

cl(i∗Z) = i∗cl(Z) ∈ H2k−2dim Y +2dim XB (X,Z).

It follows that the class map is compatible with correspondences. If X and Yare proper and smooth, and Γ ∈ CHr(X × Y ), then for every Z ∈ CHk(X), wehave

cl(Γ∗(Z)) = [Γ]∗(cl(Z)),

where [Γ]∗ : H2kB (X,Z) → H2l

B (Y,Z), l = r + k − dim X is defined by

[Γ]∗(α) = pr2∗(pr∗1α ∪ [Γ])

(cf. [86, ??]).

We will also need the cycle class for cycles on smooth quasiprojective complexalgebraic varieties. Let Z =

∑i niZi, Zi ⊂ X be such a cycle, with codim Zi =

k. In order to define[Z] ∈ H2k

B (X,Z)

we choose a smooth projective completion X of X. Then X is a Zariski openset of X and the localization exact sequence (1.1.2) shows that there exists acycle Z ∈ CHk(X) such that

Z |X = Z

and that Z is well defined up to a cycle Z ′ of X supported on X \X. Lookingat the definition of the cycle class for cycles on X, we see that the class of sucha cycle Z ′ vanishes in H2k

B (X,Z) since it vanishes in H2kB (X \ SuppZ ′,Z) and

SuppZ ′ ⊂ X \X.We thus conclude that the class [Z]|X does not depend on the choice of Z.

For similar reasons it also does not depend on the choice of compactification X.

1.2 Hodge structures

Definition 1.2.1. A weight k rational Hodge structure (L,Lp,q) consists in thedata of a Q-vector space L and a decomposition

LC := L⊗ C = ⊕p+q=kLp,q

satisfying the Hodge symmetry condition

Lp,q = Lq,p.


The Hodge filtration F ∗LC associated to such a Hodge structure is the de-creasing filtration defined by

F pLC = ⊕i≥pLi,k−i.

If X is a smooth projective variety, the k-th Betti cohomology group HkB(X,Q)

of X with rational coefficients carries a Hodge structure of weight k (cf. [86,I,7.1]). The corresponding Hodge filtration on Hk

B(X,C) is obtained using theisomorphism

HkB(X,C) = Hk(Xcl, Ω•Xan

),

where ΩiXan

is the sheaf of holomorphic i-forms on Xcl, and is induced by thenaıve filtration on the complex Ω•Xan

:

F pΩ•Xan:= Ω•≥p

Xan.

Thus the Hodge filtration is easy to define. The hard part is to prove that,letting

Hp,q(X) = F pHkB(X,C) ∩ F qHk

B(X,C), k = p + q

one has the decomposition

HkB(X,C) = ⊕p+q=kHp,q(X).

1.2.1 Polarization

Definition 1.2.2. A polarization on a weight k Hodge structure (L,Lp,q) con-sists of a nondegenerate intersection pairing ( , ) on L, which is skew-symmetricif k is odd, symmetric if k is even, and satisfying the following condition: let Hbe the Hermitian intersection pairing on LC defined by

H(a, b) = ik(a, b).

Then(i) (First Hodge-Riemann bilinear relations) The Hodge decomposition of L

is orthogonal with respect to H.(ii) (Second Hodge-Riemann bilinear relations) The restriction H|Lp,q is def-

inite, of sign (−1)p.

The interest of polarized Hodge structures lies in the following semisimplicityresult:

Theorem 1.2.3. Let (L, Lp,q) be a rational polarized Hodge structure and L′ ⊂L be a sub-Hodge structure. Then L decomposes as a direct sum of Hodgestructures:

L = L′ ⊕ L′′.


Proof. The key point is to observe that the restriction of ( , ) to L′ isnondegenerate. This follows indeed from the fact that H remains nondegenerateon each L′p,q because it is definite on each Lp,q, and from the fact that the L′p,q

are mutually perpendicular with respect to H. Having this, we observe thatbecause of the conditions (i) above, the orthogonal complement L′′ := L′⊥ ofL′ in L is a sub-Hodge structure of L. This concludes the proof since we justobserved that L′ and L′′ are supplementary.

One important application is the following Corollary 1.2.5 for which we needthe notion of a Hodge class (that we will develop in section 1.2.2).

Definition 1.2.4. Let (L,Lp,q) be a Hodge structure of weight 2k. A Hodgeclass of L is a class in L ∩ Lk,k, the intersection being taken in LC.

Corollary 1.2.5. Let φ : L → M be a morphism of rational Hodge structure,where L is polarized. Then Imφ is a sub-Hodge structure of M and L containsa sub-Hodge structure L′ such that φ|L′ is an isomorphism onto Imφ.

In particular, if β ∈ Imφ is a Hodge class, there exists a Hodge class β′ ∈ Lsuch that β = φ(β′).

Proof. The first point is obvious and does not need the polarization. LetL′′ := Ker φ. This is a sub-Hodge structure of L. By Theorem 1.2.3, there is adecomposition of L into a direct sum of Hodge structures: L = L′ ⊕ L′′. It isthen clear that φ|L′ is an isomorphism of Hodge structures onto its image.

If X is a smooth projective variety, the Hodge structures on HkB(X,Q) ad-

mit polarizations. This is a crucial difference between projective and Kahlergeometry (cf. [87]). Such polarizations are obtained as follows (cf. [86, I,7.1.2]):One chooses an ample line bundle L on X and considers the class l := c1(L) ∈H2

B(X,Q). This class satisfies the hard Lefschetz property:

ln−k∪ : HkB(X,Q) ∼= H2n−k

B (X,Q), ∀k ≤ n := dimX.

A formal consequence of this is the Lefschetz decomposition of HkB(X,Q) into

a direct sum of sub-Hodge structures:

HkB(X,Q) = ⊕k−2r≥0l

r ∪Hk−2rB (X,Q)prim,

where

Hk−2rB (X,Q)prim := Ker (Hk−2r

B (X,Q) ln−k+2r+1

→ H2n−k+2r+2B (X,Q)).

This decomposition is orthogonal with respect to the pairing

(a, b)l :=∫

X

ln−k ∪ a ∪ b

on HkB(X,Q).

The last step is to prove that up to a sign, the pairing ( , )l polarizes eachsub-Hodge structure lr ∪Hk−2r

B (X,Q)prim (cf. [86, I,6.3.2]).


1.2.2 Hodge classes

Let (L, Lp,q) be a rational Hodge structure of weight 2k. We introduced Hodgeclasses in L in Definition 1.2.4. Let X be a smooth projective complex variety.Cycle classes [Z] ∈ H2k

B (X,Q) are Hodge classes on X, that is Hodge classesfor the Hodge structure on H2k

B (X,Q) (cf. [86, I,11.3]). We will denote byHdg2k(X) the Q-vector space of degree 2k Hodge classes on X. The Hodgeconjecture states the following:

Conjecture 1.2.6. Any Hodge class α ∈ Hdg2k(X) is a linear combinationwith rational coefficients of Betti cycle classes of algebraic subvarieties of X, so

α =∑N

i=1 ai [Zi]B , ai ∈ Q.

A well-known fact is that the Hodge conjecture is true for k = 1. This isknown as the Lefschetz theorem on (1, 1)-classes (cf. [86, I,11.3]). It is also truefor degree 2n− 2 Hodge classes on smooth projective varieties of dimension n,by the degree 2 case combined with the hard Lefschetz isomorphism (4.3.23)which gives an isomorphism of Hodge structures

ln−2 : H2B(X,Q) ∼= H2n−2

B (X,Q).

This isomorphism sends cycle classes to cycle classes, by compatibility of thecycle class map with the cup-product and the intersection product, and inducesan isomorphism

Hdg2(X) ∼= Hdg2n−2(X).

Hodge classes play a crucial role in the theory of motives because of thefollowing lemma: Let X, Y be projective complex manifolds with dim X = n.Suppose that k + l = 2r is even. We apply the Kunneth decomposition. Given

α ∈ HkB(X, Q)⊗H l

B(Y, Q) ⊂ H2rB (X × Y,Q),

we can, by duality, see α as an element

α ∈ Hom(H2n−kB (X, Q),H l

B(Y, Q)).

With this terminology, we have (cf. [86, I, Lemma 11.41]):

Lemma 1.2.7. α is a Hodge class in X × Y if and only if α is a morphism ofHodge structures of bidegree (r − n, r − n).

1.2.3 Standard conjectures

Let X be a smooth projective variety of dimension n. The Kunneth decompo-sition of H∗

B(X ×X,Q) gives:

HmB (X ×X,Q) ∼=

⊕p+q=m

HpB(X,Q)⊗Hq

B(X,Q).


Poincare duality on X allows to rewrite this as

HmB (X ×X,Q) ∼=

⊕p+q=m

Hom(H2n−pB (X,Q),Hq

B(X,Q)). (1.2.6)

There are two kinds of particularly interesting Hodge classes on X × Xobtained from Lemma 1.2.7.

a) Let m = 2n and consider for each 0 ≤ q ≤ 2n the element IdHqB(X,Q)

which provides by (1.2.6) and Lemma 1.2.7 a Hodge class δq of degree 2n on X.This class is called the q-th Kunneth component of the diagonal of X. The firststandard conjecture (Kunneth’s conjecture or Conjecture C in the terminologyof [51]) is the following:

Conjecture 1.2.8. The classes δi are algebraic, that is, are classes of algebraiccycles on X ×X with rational coefficients.

b) Let L be an ample line bundle on X, and l := c1(L) ∈ H2B(X,Q). For any

integer k ≤ n, the hard Lefschetz theorem [86, I,6.2.3] says that the cup-productmap

ln−k∪ : HkB(X,Q) → H2n−k

B (X,Q)

is an isomorphism. This is clearly an isomorphism of Hodge structure. Itsinverse

(ln−k∪)−1 : H2n−kB (X,Q) → Hk

B(X,Q)

is also an isomorphism of Hodge structures, which by (1.2.6) and Lemma 1.2.7provides a Hodge class λn−k of degree 2k on X×X. The second standard conjec-ture we will consider (Lefschetz’ conjecture or Conjecture B in the terminologyof [51]) is the following:

Conjecture 1.2.9. The classes λi are algebraic, that is, are classes of algebraiccycles on X ×X with rational coefficients.

Let us state the following conjecture (which could have been stated as astandard conjecture) :

Conjecture 1.2.10. Let X be a smooth complex algebraic variety, and let Y ⊂X be a closed algebraic subset. Let Z ⊂ X be a codimension k algebraic cycle,and assume that the cohomology class [Z] ∈ H2k

B (X,Q) vanishes in H2kB (X \

Y,Q). Then there exists a codimension k cycle Z ′ on X with Q-coefficients,which is supported on Y and such that [Z ′] = [Z] in H2k

B (X,Q).

Remark 1.2.11. It is a non obvious fact that, under our assumptions, thereis a rational Hodge class β on a desingularization τ : Y → Y of Y , such that(j τ)∗β = [Z] where j is the inclusion of Y in X (see the proof of Lemma 1.2.12below). Thus Conjecture 1.2.10 is implied by the Hodge conjecture.

There is a particular numerical situation where this conjecture is proved:


Lemma 1.2.12. (cf. [94]) Conjecture 1.2.10 is satisfied by codimension k cyclesZ of X whose cohomology class vanishes away from a codimension k− 1 closedalgebraic subset Y ⊂ X.

In particular, Conjecture 1.2.10 is satisfied by codimension 2 cycles.

Proof. Indeed, if we have a codimension k cycle Z ⊂ X, whose cohomologyclass [Z] ∈ H2k

B (X,Q) vanishes on the open set X \ Y , where codim Y ≥ k − 1,then we claim that there are Hodge classes αi ∈ Hdg2k−2ci(Yi,Q), such that

[Z] =∑

i

ji∗αi,

where ji : Yi → X are projective desingularizations of the irreducible compo-nents Yi of Y , and ci := codim Yi. Indeed, we use here the fact that the “pure”part on the mixed Hodge structure on H2n−2k,B(Y,Q) is equal to the image of⊕H2n−2k,B(Yi,Q). The class [Z] comes from a class in H2n−2k,B(Y,Q), henceby Theorem 1.2.17 from a class in the pure part of H2n−2k,B(Y,Q), thus from aclass in ⊕H2n−2k,B(Yi,Q). We now use Corollary 1.2.5 to conclude that it comesfrom a Hodge class in ⊕H2n−2k,B(Yi,Q). The claim is proved. As ci ≥ k − 1for all i’s, the classes αi are cycle classes on Yi by the Lefschetz theorem on(1, 1)-classes, which concludes the proof.

The following is proved in [94]:

Proposition 1.2.13. (Voisin 2011) The Lefschetz conjecture for any X isequivalent to the conjunction of the Kunneth conjecture 1.2.8 and of Conjec-ture 1.2.10 for any X.

Proof. Let us assume that the Kunneth conjecture holds for X and Con-jecture 1.2.10 holds for any pair Y ⊂ X ′. Let i < n. Consider the Kunnethcomponent δ2n−i of ∆X , so δ2n−i ∈ Hi

B(X,Q)⊗H2n−iB (X,Q) is the class of an

algebraic cycle Z on X × X. Let Yiji→ X be a smooth complete intersection

of n− i ample hypersurfaces in X. Then the Lefschetz theorem on hyperplanesections (cf. [86, II, 1.2.2]) says that

ji∗ : HiB(Yi,Q) → H2n−i

B (X,Q)

is surjective. It follows that the class of the cycle Z vanishes on X × (X \ Yi).By Conjecture 1.2.10, there is a n-cycle Z ′ supported on X × Yi such that theclass (id, j)∗[Z ′] is equal to [Z]. Consider the morphism of Hodge structuresinduced by [Z ′]:

[Z ′]∗ : H2n−iB (X,Q) → Hi

B(Yi,Q).

Composing with the morphism ji∗ : HiB(Yi,Q) → H2n−i

B (X,Q), we get ji∗ [Z ′]∗ = IdH2n−i

B (X,Q). It follows that [Z ′]∗ is injective, and that its transpose[Z ′]∗ : Hi

B(Yi,Q) → HiB(X,Q) is surjective. We now apply [17, Proposition 8]

and induction on dimension to conclude that Lefschetz’ conjecture holds for X.


Conversely, assume Lefschetz’ conjecture holds for any smooth projectivevariety. It obviously implies the Kunneth conjecture. Let now X, Y, Z be asin Conjecture 1.2.10. Set k = codimZ, l = codim Y . Let j : Y → X be adesingularization of Y . It is known by Corollary 1.2.5 that there exists a Hodgeclass β ∈ H2k−2l

B (Y ,Q) such that j∗β = [Z]. The question is to find such aβ algebraic on Y . The argument is easier to understand if we assume thatdim X = 2dim Z and dim Y = dim X, which can always be achieved up toreplacing (X, Z) by (X × Pl1 , Z × Pl2) with dim X + l1 = 2(dim Z + l2), and Y

by Y × Pl3 , with dim Y + l3 = dim X + l1. Let thus 2n = dim X = dim Y . TheLefschetz conjecture for Y implies that the Lefschetz decomposition

H2nB (Y ,Q) = ⊕n−k≥0h

kY ∪H2n−2k

B (Y ,Q)prim,

where hY is the class of an ample hypersurface on Y , is algebraic in the sense thatthe various projectors πk on primitive factors are induced by classes of algebraiccycles on Y × Y . Let G :=

∑k(−1)kπk. G is thus induced by an algebraic cycle

of dimension 2n on Y × Y . The Hodge-Riemann bilinear relations ([86, I,6.3.2])say that the symmetric intersection pairing

< α, β >G=< α,G∗β >

is nondegenerate of a definite sign on the space Hn,n(Y ) ∩ H2nB (Y ,R) of real

cohomology classes of Hodge type (n, n) on Y , hence in particular on the spaceHdg2n(Y ) of Hodge classes on Y . It follows that the image of

j∗ : Hdg2n(Y ,Q) → Hdg2n(X,Q)

is the same as the image of

j∗ G j∗ : Hdg2n(X,Q) → Hdg2n(X,Q).

It follows that j∗Gj∗ induces an automorphism of the subspace j∗Hdg2n(Y ,Q) ⊂Hdg2n(X,Q). This automorphism is induced by an algebraic self-correspondenceof X, that is a 2n-algebraic cycle of X×X with Q-coefficients. Hence it preservesthe subspace of algebraic classes

(j∗Hdg2n(Y ,Q)) ∩H2nB (X,Q)alg ⊂ Hdg2n(X,Q).

It thus also induces an automorphism of this subspace.Recalling that [Z] = j∗β ∈ Im j∗ ∈ j∗Hdg2n(Y ,Q) ∩H2n

B (X,Q)alg, we thusconclude that [Z] = j∗((G j)∗γ) for some γ ∈ H2n

B (X,Q)alg. This concludesthe proof since (G j∗)γ is an algebraic class on Y .

Remark 1.2.14. All the (more or less) standard conjectures stated above areparticular instances of the Hodge conjecture 1.2.6. The reason why they arestated separately is the fact that the Hodge classes appearing there have auniversal character which makes them much better candidates to be classes of


algebraic cycles : they are absolute Hodge classes (cf. [29], [90]). This meansthat they have certain special properties satisfied by cycle classes, and the moststriking one from the viewpoint adopted there is the following: Grothendieck’stheorem [42] says that the cohomology with complex coefficients H2k

B (X,C) ofa complex algebraic variety can be computed as algebraic de Rham cohomology,that is via algebraic differential forms. It follows that if X is defined over a fieldK ⊂ C then this cohomology group is also defined over K. This K-structurehas nothing to do with the Betti Q-structure of H2k

B (X,C). One property ofabsolute Hodge classes of degree 2k is that, after multiplication by (2iπ)k, theybecome defined over a finite extension of K.

1.2.4 Mixed Hodge structures

Definition 1.2.15. A rational (real) mixed Hodge structure of weight n is givenby a Q-vector space (R-vector space) H equipped with an increasing filtrationWiH called the weight filtration, and a decreasing filtration on HC := H ⊗ C,called the Hodge filtration F kHC. The induced Hodge filtration on each GrW

i His required to equip GrW

i H with a Hodge structure of weight n + i.

Naturally, these filtrations are also required to satisfy F iH = 0 for suffi-ciently large i, F iH = H for sufficiently small i, and similarly WiH = 0 forsufficiently small i, WiH = H for sufficiently large i.

Equivalently, for each i, k, we must have

GrWi HC = F kGrW

i HC ⊕ Fn+i−k+1GrWi HC,

whereF kGrW

i HC = Im(F kHC ∩WiHC → GrWi HC).

Remark 1.2.16. All mixed Hodge structures in Deligne’s sense have weight0. However, we find that the mixed Hodge structures coming from geometryhave a natural weight (indicated by geometry), which is the one we attribute tothe pure part. Thus there is in these notes a shift of the indices for the weightfiltration with respect to standard terminology.

Note that with these notations, one can define the twist L′ = L(r) of amixed Hodge structure L. It is the mixed Hodge structure of weight n − 2rwith the same underlying rational vector space as L obtained by deciding thatWiL

′ = W2r+iL and F pL′C = F r+pLC.We have the obvious notion of a morphism of mixed Hodge structures. A

morphism α of filtered vector spaces (U,F ) and (V, G) is said to be strict if

Imα ∩GpV = α(F pU).

It is an elementary fact (see [86, II, 7.3.1]) that the morphisms of rational Hodgestructures are strict for the Hodge filtration. This result extends to mixed Hodgestructures:


Theorem 1.2.17. (Deligne, [25]) The morphisms

α : (H, W,F ) → (H ′,W ′, F ′)

of (rational or real) mixed Hodge structures are strict for the filtrations W andF .

This result follows from the following fact for which we refer to [25] or [86,??].

Lemma 1.2.18. Let (H, W,F ) be a mixed Hodge structure. There exists adecomposition as a direct sum

HC = ⊕p,qHp,q, (1.2.7)

with Hp,q ⊂ F pHC ∩Wp+q−nHC, such that under the projection Wp+q−nHC →GrW

p+q−nHC, Hp,q can be identified with

Hp,q(GrWp+q−nHC) := F pGrW

p+q−nHC ∩ F qGrWp+q−nHC.

More generally, we have

WiHC = ⊕p+q≤n+iHp,q (1.2.8)

F iHC = ⊕p≥iHp,q. (1.2.9)

This decomposition is respected by the morphisms of mixed Hodge structures.

Proof of theorem 1.2.17. Indeed, if l′ ∈ α(HC) ∩W ′iH

′, let us write l′ =α(l) and decompose l =

∑lp,q as in (1.2.7). Then α(l) admits the decomposition

α(l) =∑

α(lp,q) with α(lp,q) ∈ H ′p,q. But as l′ ∈ W ′iH

′C, we have α(lp,q) = 0

for p + q > n + i. Thus,

l′ = α(∑

p+q≤n+i

lp,q) ∈ α(WiHC).

Therefore,Im αC ∩W ′

iH′C = αC(WiHC).

It is then easy to see that this still holds when C is replaced by R or Q.The same argument shows that α is also strict for the filtration F .

1.2.5 Coniveau

Definition 1.2.19. A weight k Hodge structure (L,Lp,q) has (Hodge) coniveauc ≤ k

2 if the Hodge decomposition of LC takes the form

LC = Lk−c,c ⊕ Lk−c−1,c+1 ⊕ . . .⊕ Lc,k−c

with Lk−c,c 6= 0.


If L has coniveau ≥ r, we can define a Hodge structure L(r) of weight k−2rwith the same underlying Q-vector space as L, and Hodge decomposition

L(r)p,q = Lp+r,q+r. (1.2.10)

A fundamental result is the following (cf. [41]):

Theorem 1.2.20. If X is a smooth complex projective variety and Y ⊂ X isa closed algebraic subset of codimension c, then Ker (j∗ : Hk

B(X,Q) → HkB(X \

Y,Q)), where j : X \ Y → X is the inclusion map, is a sub-Hodge structure ofconiveau ≥ c of Hk

B(X,Q).

Proof. Choose a desingularization τ : Y → Y , and assume that Y has purecomplex dimension n− c.

We need the fact that morphisms of mixed Hodge structures are strict for theweight filtration (cf. Theorem 1.2.17). We already know (via Poincare dualityfor open varieties) that the kernel of j∗ is the same as the image of

i∗ : H2n−k,B(Y,Q) → H2n−k,B(X,Q)DX∼= Hk

B(X,Q),

where DX is the poincare duality isomorphism. There is a mixed Hodge struc-ture on both sides, of respective weights k − 2c, k. The composition is a mor-phism of mixed Hodge structures of bidegree (c, c), with a pure Hodge structureon the right (cf. [25]). Thus by Theorem 1.2.17, its image is the same as theimage of the map

i∗ : W0H2n−k,B(Y,Q) → H2n−k,B(X,Q) ∼= HkB(X,Q).

But by construction, the pure part W0H2n−k,B(Y,Q) of the mixed Hodge struc-ture on H2n−k,B(Y,Q) coincides with the image of the map

τ∗ : H2n−k,B(Y ,Q) → H2n−k,B(Y,Q),

where on the left we have a pure Hodge structure of weight k − 2c. Thatconcludes the proof, because, applying Poincare duality on Y , we proved thatKer j∗ = Im (i τ)∗ : Hk−2c

B (Y ,Q) → HkB(X,Q) and this morphism is a mor-

phism of Hodge structures of bidegree (c, c).

The conjecture below, due to Grothendieck, proposes a characterization ofcohomology classes supported on a subvariety of codimension ≥ c (correctingthe original Hodge conjecture):

Conjecture 1.2.21. (Generalized Hodge conjecture, Grothendieck [41]) LetL ⊂ Hk

B(X,Q) be a rational sub-Hodge structure of Hodge coniveau ≥ c. Thenthere exists a closed algebraic subset Z ⊂ X of codimension c such that L van-ishes under the restriction map Hk

B(X,Q) → HkB(U,Q), where U := X \ Z.

Let us explain the link between the “standard” Hodge conjecture 1.2.6 andthe generalized Hodge conjecture:


The Hodge conjecture implies the generalized Hodge conjecture in two par-ticular cases. The most obvious case occurs when k = 2c. In this case we haveLC = Lc,c and L consists of Hodge classes. The Hodge conjecture providescodimension c cycles Z1, . . . , ZN of X such that

L = 〈[Z1], . . . , [ZN ]〉 ⊗Q.

But then L vanishes on X \ (SupZ1 ∪ · · · ∪ Sup ZN ), as required.The next, much more sophisticated, case is that in which k = 2c+1, so that

LC = Lc+1,c ⊕ Lc,c+1. (1.2.11)

We follow here [70, ??]. In this case, referring to (1.2.10), L′ = L(c) is apolarized Hodge structure of weight 1. We get such Hodge structures on thefirst cohomology groups of curves, though not every Hodge structure of weight1 is actually a Hodge structure of a curve. However,

Lemma 1.2.22. Any polarized Hodge structure of weight 1 arises as H1B(A, Q)

for some abelian variety A.

The key point is the existence of polarization (arising from the intersectionform), but without polarizations, the lemma remains true with “abelian vari-eties” replaced by “complex tori”. The lemma is a reformulation of the factthat the category of weight 1 rational polarized Hodge structures is the same asthe category of abelian varieties up to isogeny (see [86, I,Section 7.2.2]).

Using this lemma, we shall prove

Theorem 1.2.23. (cf. [70]) The Hodge conjecture implies Conjecture 1.2.21when k = 2c + 1 as in (1.2.11).

Proof. We start with a Hodge substructure L ⊂ H2c+1B (X,Q) of coniveau

c. It is polarized, because the Hodge structure on H2c+1B (X,Q) is polarized

(note however that the polarizations are not canonical). There exists then apolarized Hodge structure L′ of weight 1 and a morphism φ : L′ ∼= L of bidegree(c, c). By Lemma 1.2.22 we may assume L′ = H1

B(A, Q) as Hodge structures.Having an abelian variety A and a morphism of Hodge structures H1

B(A, Q) →H2c+1

B (X, Q), we can choose a curve C that is a complete intersection of amplehypersurfaces in A. Then, by the Lefschetz theorem on hyperplane sections [86,II,1.2.2], there is a monomorphism

i∗ : H1B(A, Q) → H1

B(C, Q), (1.2.12)

which is a morphism of Hodge structures of pure weight 1. Since the categoryof polarized Hodge structures of weight 1 is semisimple, (1.2.12) always splits,so H1

B(A, Q) is a direct summand, as a Hodge substructure, of H1B(C, Q). We

thus get a morphism of Hodge structures ψ : H1B(C, Q) → H2c+1

B (X, Q) ofbidegree (c, c). By Lemma 1.2.7, ψ determines a Hodge class ψ of degree 2c + 2


on C×X. If the usual Hodge conjecture is true on C×X, then this class is theclass of a codimension c + 1 cycle Z =

∑i aiZi of C ×X:

ψ = [Z] =∑

ai[Zi], ai ∈ Q.

The resulting correspondence induces maps

H1B(SupZ, Q)

p∗1←−H1B(C, Q)

∑i aip2,i∗

y[Z]∗

H2c+1B (X, Q)

In truth, one needs to desingularize the components Zi of Z and to replace thesupport Sup Z of Z by

⊔Zi. Then Im([Z]∗) = Im ψ = L vanishes away from

p2(Sup Z) which is of codimension c. Thus, L satisfies the generalized Hodgeconjecture.

The above argument shows that the usual Hodge conjecture for varietieswhich are products with a curve implies generalized Hodge conjecture for weightn = 2c + 1 and coniveau c. In order to deduce by a similar argument thegeneralized Hodge conjecture from the standard Hodge conjecture, there is animportant missing point, namely, the adequate generalization of Lemma 1.2.22.This leads to the following question:

Question 1.2.24. Given a weight k Hodge structure L ⊂ HkB(X, Q) of coniveau

r, so thatLC = Lk−r,r ⊕ · · · ⊕ Lr,k−r,

consider the weight k − 2r Hodge structure L′ = L(r) which has the same un-derlying lattice as L and Hodge decomposition

L′p,q = Lp+r,q+r.

Does there exist a smooth projective variety Y admitting L′ as a Hodge sub-structure of Hk−2r

B (Y, Q)?

Chapter 2

Decomposition of thediagonal

In this chapter, we explain the method initiated by Bloch and Srinivas, anddeveloped later on independently by Lewis, Schoen, Laterveer and Paranjape,which leads to statements of the following shape (cf. Theorem 2.2.3):

If a smooth projective variety has trivial Chow groups of dimension ≤ c− 1,then its cohomology has geometric coniveau ≥ c.

This statement is a vast generalization of Mumford’s theorem [61]. A majoropen problem is the converse to this statement.

It turns out that statements of this kind are a consequence of a general spread-ing principle for rational equivalence (cf. Theorem 2.1.1): Consider a smoothprojective family X → B and a cycle Z → B, everything defined over C.

If at the very general point b ∈ B, the restricted cycle Zb ⊂ Xb is rationallyequivalent to 0, there exists a dense Zariski open set U ⊂ B and an integer Nsuch that NZU is rationally equivalent to 0 on XU .

We will spell-out the consequences of this principle, as the generalized de-composition of the diagonal, and state other spreading principles which will beused in subsequent chapters.

2.1 A general principle

The following result is essentially due to Bloch-Srinivas [14]. Let f : X → Ybe a smooth projective morphism, where X and Y are smooth (for simplicity),and let Z ∈ CHk(X) be a cycle in X. Consider the following property:

(*)There exists a subvariety X ′ ⊂ X such that for every y ∈ Y , the cycleZy := Z|Xy

:= j∗yZ ∈ CHk(Xy) vanishes in CHk(Xy − X ′y), where jy is the

inclusion of the fiber Xy = f−1(y) into X.

39

40 CHAPTER 2. DECOMPOSITION OF THE DIAGONAL

Theorem 2.1.1. If Z satisfies the property (*), then there exists an integerm > 0, and a cycle Z ′ supported in X ′, such that we have the equality

mZ = Z ′ + Z ′′ in CHk(X), (2.1.1)

where Z ′′ is a cycle supported in XY ′ := f−1(Y ′) for some proper closed alge-braic subset Y ′ ⊂ Y .

We refer to [14] or to [86, ??] for the proof of this Theorem. Note thatthe statement given here is slightly different from the one given in [14], thereason being the fact that we work over C, which is very big, and contains inparticular any finitely generated field over Q or the algebraic closure of suchfield. The statement above would be (expectedly) wrong if we were workingover a countable field like Q and considering only the closed points y ∈ Y (Q).A conjecture of Beilinson says indeed that looking at cycles defined over Q on avariety defined over Q does not allow to conclude anything on the Chow groupsof the corresponding variety over C. What makes this expectation possible isthe following basic fact:

Lemma 2.1.2. Let f : X → Y be a projective fibration, where X and Y aresmooth, and let Z ∈ CHk(X) be a cycle in X. Then the set of points y ∈ Y suchthat the restricted cycle Zy is rationally equivalent to 0 is a countable union ofclosed algebraic subsets of Y .

Assuming everything defined over Q, this countable union could exhaustY (Q) without saying anything over what happens over the very general pointor the geometric generic point.

Remark 2.1.3. Note also that Lemma 2.1.2 shows that in Theorem 2.1.1,we could have replaced our assumption that Zy := Z|Xy

:= j∗yZ ∈ CHk(Xy)vanishes in CHk(Xy − X ′

y) for all y ∈ Y by the condition that that Zy :=Z|Xy

:= j∗yZ ∈ CHk(Xy) vanishes in CHk(Xy − X ′y) for a very general point

y of Y , that is for any point of Y in the complement of a countable union ofproper closed algebraic subsets of Y . Indeed, by a Baire category argument andLemma 2.1.2, the two assumptions are equivalent.

We will apply this principle in section 4.3, in the following rather simpleform: First of all we assume the variety X ′ above is empty. Secondly, we justconsider the cohomological version of (2.1.1):

Corollary 2.1.4. Let π : X → Y be a smooth morphism, where X and Y aresmooth, and let Z be a codimension k algebraic cycle on X. Assume that therestrictions Z|Xy

vanish in CHk(Xy) for any y ∈ Y . Then there is a nonzerointeger m and a proper closed algebraic subset Y ′ ⊂ Y such that

m[Z] = 0 in H2kB (X \XY ′ ,Z).

Equivalently [Z] = 0 in H2kB (X \XY ′ ,Q).

2.1. A GENERAL PRINCIPLE 41

It is important to point out that this corollary holds only for rational equiv-alence, and not for most weaker equivalence relation. For example, it is com-pletely wrong for algebraic equivalence, where a cycle Z ∈ Zk(X) is said tobe algebraically equivalent to 0 if there exists a smooth curve C, a 0-cyclez ∈ Z0(C) homologous to 0, and a correspondence Γ ∈ Zk(C × X) such thatZ = Γ∗(z). If one takes a smooth projective family of curves C → B, and afamily of zero cycles homologous to zero in the fibers Z ⊂ C, the cycles are alge-braically equivalent to 0 in the closed fibers (and also in the geometric genericfiber) but its class [Z] ∈ H2

B(C,Q) is nonzero in general (and even does notvanish over any Zariski open set of B): It is computed by the theory of the classof a normal function [86, ??]. What happens here is the following : If we replacein Corollary 2.1.4 rational equivalence by algebraic equivalence, our assumptionis (by a countability argument) equivalent to the fact that the restriction ofthe cycle Z to the geometric generic fiber (which is a variety Xη defined overC(Y )) is algebraically equivalent to 0. But this does not imply of course thatthere exists a Zariski open set Y 0 of Y such that, denoting X0 := π−1(Y 0), therestriction Z|X 0 is up to torsion algebraically equivalent to 0 over C, for thefollowing reason: there is a curve Cη defined over C(Y ) and a 0-cycle z homol-ogous to 0 on Cη, such that Zη is as above the image of z under a codimensionk correspondence Γ between Cη and Xη. This correspondence can be spreadout over a finite cover of a Zariski open set of Y , and this can be done as wellfor the curve Cη and the cycle z, giving rise to families over a finite cover of aZariski dense open set Y 0 of the base. The problem is that the spread-out curveC is not in general isotrivial on Y 0. Even if it was isotrivial, hence isomorphicto C0× Y 0 (maybe up to passing to a finite etale cover of Y 0), the codimension1 cycle z would spread to a codimension 1 cycle Z ′ on C0 × Y 0 which mightnot be cohomologous to 0 over any Zariski open set of Y 0, if g(C0) > 0. Indeedwe just know that it it cohomologous to 0 on the fibers of pr2 : C0 × Y 0 → Y 0,but when g(C0) > 0, the Kunneth decomposition of H2

B(C0 × Y 0,Q) involves anon trivial term H1

B(C0,Q)⊗H1B(Y 0,Q) which in general won’t vanish on any

Zariski open set of Y 0. This last argument, which can be stated in general as aLeray spectral sequence argument for the morphism C → Y 0, makes the wholedifference between rational and algebraic equivalence.

In order to state different applications of Theorem 2.1.1, we consider nowthe case where X = X1 × Y , f is the second projection, and X ′ = X ′

1 × Y . Wethen obtain the following result.

Corollary 2.1.5. Let Γ ∈ CHk(X1 × Y ), and assume that for every y ∈ Y ,the cycle Γ∗(y) ∈ CHk(X1) restricts to zero in CHk(X1 −X ′

1). Then we have adecomposition

mZ = Z ′ + Z ′′ ∈ CHk(X1 × Y ),

where Z ′ is supported in X ′1 × Y and Z ′′ is supported in X1 × Y ′, for a proper

closed Y ′ ⊂ Y .

Remark 2.1.6. Our assumption is equivalent, by the localization exact se-quence (1.1.2), to the fact that the cycle Γ∗(y) is rationally equivalent to a cycle


supported on X ′1.

Applying this result to the diagonal of a smooth projective variety Y , we getthe famous Bloch-Srinivas decomposition of the diagonal:

Theorem 2.1.7. (Bloch-Srinivas [14]) If Y is a smooth projective variety suchthat CH0(Y ) is supported on some closed algebraic subset W ⊂ Y , there is anequality in CHd(Y × Y ), d = dimY :

N∆Y = Z1 + Z2, (2.1.2)

where N is a nonzero integer and Z1, Z2 are codimension d cycles with

SuppZ1 ⊂ D × Y, D Y, SuppZ2 ⊂ Y ×W.

Remark 2.1.8. The meaning of the integer N appearing above is discussed inthe paper [93] (see also Section 5.3). If dim Y = 3 and dimW ≤ 1, N controlsfor example the torsion in H∗

B(Y,Z), hence it cannot be in general set equal to1.

Although the cohomological version of the statement above may seem muchweaker, it leads in practice to a number of applications. We thus state it sepa-rately as the cohomological Bloch-Srinivas decomposition of the diagonal :

Corollary 2.1.9. (Cohomological decomposition of the diagonal) If Y is asmooth projective variety such that CH0(Y ) is supported on some closed alge-braic subset W ⊂ Y , there is an equality of cycle classes in H2d

B (Y × Y,Q), d =dimY :

N [∆Y ] = [Z1] + [Z2], (2.1.3)


SuppZ1 ⊂ T × Y, T Y, SuppZ2 ⊂ Y ×W.

2.1.1 Mumford’s theorem

The following is a generalization of Mumford’s theorem [61] which concernedsurfaces. The proof we give below is due to Bloch and Srinivas:

Theorem 2.1.10. (Generalized Mumford’s theorem.) Let X be smooth projec-tive of dimension n. If there exists a closed algebraic subset j : X ′ → X suchthat dim X ′ < r and the map

j∗ : CH0(X ′) → CH0(X)

is surjective, then H0(X, ΩkX) = 0 for k ≥ r.


Proof. We use the decomposition (2.1.3). We thus have equality of thecorresponding cohomology classes:

m[∆X ] = [Z ′] + [Z ′′] ∈ H2nB (X ×X,Z),

where m is a nonzero integer and Z ′, Z ′′ are codimension n cycles of X × Xwith

Supp Z ′ ⊂ T ×X, T X, SuppZ ′′ ⊂ X ×X ′.

We can view these cohomology classes, or rather their Kunneth components ofadequate degree, as morphisms of Hodge structures

m[∆X ]∗, [Z ′]∗, [Z ′′]∗ : HkB(X,Z) → Hk

B(X,Z)

(cf. [86, I,11.3.3]). Then we have the equality

m[∆X ]∗ = [Z ′]∗ + [Z ′′]∗ ∈ Hom(HkB(X,Z),Hk

B(X,Z)). (2.1.4)

Now, the morphism [∆X ]∗ is equal to the identity, by the fact that ∆X isthe diagonal. Thus, (2.1.4) gives

mId = [Z ′]∗ + [Z ′′]∗ ∈ Hom(HkB(X,Z),Hk

B(X,Z)).

In particular, for η ∈ H0(X, ΩkX) ⊂ Hk

B(X,C), we have

mη = [Z ′]∗η + [Z ′′]∗η ∈ HkB(X,C). (2.1.5)

Now let l : T → X be a desingularisation of T . As the cycle Z ′′ is supported inT ×X, it comes from a cycle Z ′′ of T ×X, Z ′′ = (l, Id)∗(Z ′′), so we get

cl(Z ′′) = (l, Id)∗(cl(Z ′′)).

Recall that if α ∈ H2nB (X ×X,Z), the corresponding morphism

α∗k : HkB(X,Z) → Hk

B(X,Z)

is defined by

α∗k(β) = p1∗(p∗2(β) ∪ α). (2.1.6)

Formula (2.1.6) then shows that

[Z ′′]∗ = l∗ [Z ′′]∗. (2.1.7)

Similarly, let j : X ′ → X be a desingularization of X ′, and let Z ′ be a cycle ofX × X ′ such that (Id, j)∗(Z ′) = Z ′. Then the formula (2.1.6) shows that

[Z ′]∗ = [Z ′]∗ j∗. (2.1.8)

By the equations (2.1.7) and (2.1.8), (2.1.5) now gives

mη = [Z ′]∗ j∗η + l∗ [Z ′′]∗η. (2.1.9)


But as dim X ′ < r, for k ≥ r, we have

j∗η = 0 in H0(X ′, ΩkX′).

Thus, we have [Z ′]∗ j∗η = 0 and

mη = l∗ [Z ′′]∗η. (2.1.10)

Moreover, as dim T < dim X, the morphism of Hodge structures l∗ is of bidegree(s, s) with s = codim T > 0, so the intersection of its image with H0(X, Ωk

X) =Hk,0(X) is reduced to 0. Thus, the equality (2.1.10) implies that η = 0 for

η ∈ H0(X, ΩkX), k ≥ r.

2.1.2 Applications

Let us give two further applications which use only the cohomological versionof the decomposition of the diagonal (Corollary 2.1.9):

Theorem 2.1.11. (Bloch-Srinivas [14]) Let X be a smooth complex projectivevariety such that there exists a closed algebraic subset j : X ′ → X, of dimension≤ 3, such that the map

j∗ : CH0(X ′) → CH0(X)

is surjective. Then the Hodge conjecture holds for classes of degree 4.

This result was originally proven by Conte and Murre [21] in the case whereX is a 4-dimensional variety covered by rational curves. Such a variety X satis-fies the hypothesis, since every point x is contained in a rational curve Cx whosenormalisation is isomorphic to P1. By the definition of rational equivalence, allthe points y ∈ Cx are rationally equivalent in Cx, so also in X, and if X ′ is anample hypersurface of X, then Cx intersects X ′ and x is rationally equivalentin X to any point of X ′ ∩ Cx.

Proof of proposition 2.1.11. Applying corollary 2.1.9, we see that thereexists a proper subset T ⊂ X, which we may assume to be of codimension 1, andn-dimensional cycles Z ′ and Z ′′ supported in T ×X and X ×X ′ respectively,such that

m[∆X ] = [Z ′] + [Z ′′] in H2n(X ×X,Z) (2.1.11)

for some non-zero integer m. Let k : T → X and j : X ′ → X be desingu-larizations of T and X ′ respectively. The decomposition (2.1.11) then yields


the equalities of the morphisms of Hodge structure associated to the Kunnethcomponents of type (2n− 4, 4) of these three Hodge classes:

m[∆X ]∗ = [Z ′]∗ + [Z ′′]∗ : H4B(X,Z) → H4

B(X,Z).

Now, let Z ′ ⊂ T ×X be a cycle of codimension n such that

(k, Id)∗Z ′ = Z ′,

and let Z ′′ ⊂ X × X ′ be a cycle of codimension n such that

(Id, j)∗Z ′′ = Z ′′.

We havecl(Z ′) = (k, Id)∗cl(Z ′), cl(Z ′′) = (Id, j)∗cl(Z ′′),

and it follows that for every α ∈ H4B(X,Z), we have

[Z ′]∗α = k∗[Z ′]∗α, [Z ′′]∗α = [Z ′′]∗(j∗α).

But as X ′ is of dimension ≤ 3, the rational Hodge conjecture holds for X ′ inevery degree. (Indeed, it holds for classes of degree 2 by the Lefschetz theoremon the classes of type (1, 1), and for classes of degree 4 by the argument given inSection 1.2.2. Moreover, the Hodge conjecture is satisfied for classes of degree0 and 2n for every smooth projective variety of dimension n.)

If α ∈ H4B(X,Q) ∩ H2,2(X) is a rational Hodge class, the classes j∗α ∈

H4B(X ′,Q) ∩ H2,2(X ′) and [Z ′]∗α ∈ H2

B(T ,Q) ∩ H1,1(T ) are thus classes ofalgebraic cycles of X ′ and T respectively. The relation

m[∆X ]∗α = mα = k∗[Z ′]∗α + [Z ′′]∗(j∗α)

and the compatibility of the cycle class map with correspondences then showthat α is also the class of an algebraic cycle with rational coefficients.

The second application is already contained in the proof of the generalizedMumford theorem 2.1.10. It is important enough however to justify a separatestatement. Using the notion of coniveau introduced in section 1.2.5, we canrestate Theorem 2.1.10 by saying that if X has its group CH0(X) supported onsome X ′ ⊂ X of dimension < r, then the coniveau of the Hodge structures onHk

B(X,Q) is at least 1 for k ≥ r. Recall that the generalized Hodge conjecturethen predicts that Hk

B(X,Q) is supported on a proper algebraic subset of X.This is indeed what the proof of Theorem 2.1.10 shows.

Theorem 2.1.12. Let X be smooth projective of dimension n. If there existsa closed algebraic subset j : X ′ → X such that dim X ′ < r and the map

j∗ : CH0(X ′) → CH0(X)

is surjective, then for k > r, the cohomology groups HkB(X,Q) vanish on a dense

Zariski open set of X. In other words, H∗>rB (X,Q) has geometric coniveau ≥ 1.


Proof. We start as in the proof of Theorem 2.1.10 and write again theequality for any η ∈ Hk

B(X,Q).

mη = [Z ′]∗ j∗η + l∗ [Z ′′]∗η.

of ( 2.1.9), where l : T → X be a desingularization of T (which we may assumeto be a divisor of X and j : X ′ → X is a desingularization of X ′.

By definition, l∗ [Z ′′]∗η is of geometric coniveau ≥ 1. It thus suffices toprove that [Z ′]∗ j∗η is also of geometric coniveau ≥ 1 if deg η > r := dimX ′.This is because the degree k > r class j∗η has geometric coniveau ≥ 1 on X ′ bythe Lefschetz isomorphism (4.3.23) applied to X ′.

Remark 2.1.13. Because the integer m cannot in general be set equal to 1, (cf.[93]), the proof above only works for cohomology with rational coefficients. How-ever, the Bloch-Kato conjecture proved by Rost and Voevodsky ([82]) impliesthat the result holds as well for cohomology with integral coefficients. Indeed,the Bloch-Kato conjecture implies (cf. [14], [22], [5] and Section 5.2.2) that theBloch-Ogus sheaves of Z-modules Hk(Z) on XZar associated to the presheavesU 7→ Hk

B(U,Z) have no torsion.

2.2 Varieties with small Chow groups

2.2.1 Generalized decomposition of the diagonal

The Bloch-Srinivas decomposition of the diagonal has been generalized by Paran-jape [67] and Laterveer [56] under triviality assumptions on Chow groups ofsmall dimension.

Theorem 2.2.1. Assume that for k < c, the maps

cl : CHk(X)⊗Q→ H2n−2kB (X,Q)

are injective. Then there exists a decomposition

m∆X = Z0 + . . . Zc−1 + Z ′ ∈ CHn(X ×X), (2.2.12)

where m 6= 0 is an integer, Zi is supported in W ′i × Wi with dim Wi = i and

dim W ′i = n−i, and Z ′ is supported in T×X, where T ⊂ X is a closed algebraic

subset of codimension ≥ c.

Theorem 2.1.7 is in fact the case c = 1 of this theorem.

Proof of theorem 2.2.1. We use induction on c. The case c = 1 wasalready considered. We may thus assume that we have a decomposition

m∆X = Z0 + . . . + Zc−2 + Z ′ ∈ CHn(X ×X), (2.2.13)

2.2. VARIETIES WITH SMALL CHOW GROUPS 47

where m 6= 0 is an integer, Zi is supported in W ′i × Wi with dimWi = i and

dim W ′i = n−i, and Z ′ is supported in T×X, where T ⊂ X is a closed algebraic

subset of codimension ≥ c− 1. We may assume that T is of codimension c− 1.Let τ : T → X be a desingularization. Let Z ′ ⊂ T × X be a cycle such that(τ, Id)∗Z ′ = Z ′. The cycle Z ′ ⊂ T ×X is of codimension n− c + 1 and inducesa morphism Z ′∗ : CH0(T ) → CHc−1(X). Now, we know by assumption that thekernel of cl : CHc−1(X) → H2n−2k0

B (X,Z) is torsion. Thus, the map Z ′∗ mapsCH0(T )hom to the torsion of CHc−1(X). By a Baire countability argument, itfollows that there exists an integer M such that MZ ′∗ = 0 on CH0(T )hom.

For each component Ti of T , choose a point ti ∈ Ti as above, and let Wi =Z ′∗(ti). The cycle

Z ′′ = M(Z ′ −∑

i

Ti ×Wi) ⊂ T ×X

then satisfies the property that its restriction to each t×X is trivial in CHc−1(X).We then apply Theorem 2.1.1 to conclude that there exists a cycle Z ′′ ⊂ T ′×X,where T ′ ⊂ T is a proper closed algebraic subset, and an integer M ′, such that

M ′M(Z ′ −∑

i

Ti ×Wi) = Z ′′ in CHn(T ×X).

Setting T ′ = τ(T ′) and Z ′′ = τ∗Z ′′, we obtain

M ′M(Z ′ −∑

i

Ti ×Wi) = Z ′′ in CHn(X ×X),

with Z ′′ supported on T ′ ×X and codim T ′ ≥ c. Combining this equality with(2.2.13), we obtain the desired decomposition.

Theorem 2.2.1 can be used to give a proof of the following result due toLewis [57]:

Theorem 2.2.2. (Lewis 1995) Let X be a smooth projective complex varietysuch that the cycle class map cl : CHi(X)Q → H2i(X,Q) is injective for alli ≥ 0. Then H2i+1(X,Q) = 0 for all i ≥ 0 and the cycle class map cl :CHi(X)Q → H2i(X,Q) is an isomorphism for all i ≥ 0.

Proof. Indeed, by Theorem 2.2.1, we have under these assumptions a com-plete decomposition of the diagonal

∆X =∑

i,j

nijWi ×Wj in CHn(X ×X)Q.

The corresponding cohomological decomposition writes

[∆X ] =∑

i,j

nijpr∗1 [Wi] ∪ pr∗2 [Wj ] in H2n(X ×X)Q.


For any α ∈ H∗(X,Q) we thus get

α = [∆X ]∗α =∑

i,j

nij < α, [Wi] > [Wj ],

which shows that the rational cohomology of X is generated by classes of alge-braic cycles.

2.2.2 Generalized Bloch conjecture

The main application of the generalized decomposition of the diagonal is thefollowing result.

Theorem 2.2.3. Let X be a smooth projective variety of dimension m. Assumethat the cycle class map


is injective for i ≤ c − 1. Then we have Hp,q(X) = 0 for p 6= q and p < c (orq < c).

More precisely, the Hodge structures on HkB(X,Q)⊥ alg are all of Hodge

coniveau ≥ c; in fact they even are of geometric coniveau ≥ c, hence theysatisfy the generalized Hodge conjecture 1.2.21 for coniveau c.

Here HkB(X,Q)⊥ alg denotes the transcendental part of the cohomology, de-

fined as the subspace of HkB(X,Q) consisting of classes orthogonal to all classes

of algebraic cycles (of degree 2n − k). Of course it is different from HkB(X,Q)

only if k is even.

Proof of Theorem 2.2.3. Let us write the cohomological generalizeddecomposition of the diagonal of theorem 2.2.1:

m[∆X ] = [Z0] + . . . + [Zc−1] + [Z ′] ∈ H2n(X ×X,Q).

By hypothesis, the Zi are of the form

Zi =∑

j

ni,jW′i,j ×Wi,j , (2.2.14)

where the Wi,j (resp. W ′i,j) are irreducible components of Wi (resp. W ′

i ).Moreover, Z ′ is supported in T × X with codim T ≥ c. Let l : T → X be adesingularization of T , and let Z ′ in CHn(T×X) be such that (l, Id)∗(Z ′) = Z ′.The above decomposition gives a decomposition of the corresponding morphismsof Hodge structures for every k:

m[∆X ]∗ = mId = [Z0]∗ + . . . [Zk]∗ + [Z ′]∗ : HkB(X,Q) → Hk

B(X,Q).


Now, by (2.2.14), we clearly have

[Zi]∗(α) =∑

j

ni,j < α, [Wi,j ] > [W ′i,j ], (2.2.15)

where < , > is the intersection form on H∗B(X,Q). In particular, we have

[Zi]∗(α) = 0 ∀α ∈ Hp,q(X), p 6= q.

For α satisfying this hypothesis, we thus have

mα = [Z ′]∗(α) = l∗([Z ′]∗(α)). (2.2.16)

Thus, if α ∈ Hp,q(X) with p 6= q, we have

mα ∈ Im l∗ ∩Hp,q(X).

Now, as we may assume codimT = c, the Gysin morphism

l∗ : Hp+q−2cB (T ,Z) → Hp+q

B (X,Z)

is a morphism of Hodge structures of bidegree (c, c), so that

Im l∗ ∩Hp,q(X) = 0, q ≤ c− 1.

Hence we have mα = 0 for

α ∈ Hp,q(X), p 6= q, q ≤ c− 1.

This proves the first statement.For the second statement, formula (2.2.15) shows more precisely that [Zi]∗(α) =

0 for α ∈ HkB(X,Q)⊥ alg. For such α, formula (2.2.16) holds and shows that

α ∈ Im l∗, which completes the proof, since Im l∗ vanishes away from T , and Thas codimension ≥ c.

The major open problem in the theory of algebraic cycles, which by the abovetheorem would also solve many instances of the generalized Hodge conjecture,is the following conjecture relating the Hodge coniveau and Chow groups. Thisconverse of Theorem 2.2.3 is a vast generalization of Bloch conjecture for surfaces[13].

Conjecture 2.2.4. (cf. [86, II, 11.2.2]) Let X be a smooth projective varietyof dimension m satisfying the condition Hp,q(X) = 0 for p 6= q and p < c (orq < c). Then for any integer i ≤ c− 1, the cycle map


is injective.


Remark 2.2.5. This conjecture can be split into two parts. Indeed, as weproved in the previous section, if the conclusion of the conjecture holds true,then the cohomology of X (or rather its transcendental part) is supported on aclosed algebraic subset of codimension ≥ c. This is predicted by the generalizedHodge conjecture 1.2.21.

If we assume Conjecture 1.2.21, the Bloch conjecture can be reformulatedas follows, without any mention of Hodge structures:

Conjecture 2.2.6. If the transcendental cohomology of X is supported on aclosed algebraic subset of codimension ≥ c, then for any integer i ≤ c − 1, thecycle map


is injective.

In the following section, we sketch the ideas of Kimura, which lead to a proofof Conjecture 2.2.6 for varieties dominated by products of curves. In a ratherdifferent geometric setting, we will prove in section 3.3 Conjecture 2.2.6 for verygeneral complete intersections in a variety with trivial Chow groups, assumingthe Lefschetz standard conjecture.

2.2.3 Nilpotence conjecture and Kimura’s theorem

Let X be smooth projective, and Γ ⊂ X × X a correspondence. Recall thatcorrespondences between smooth projective varieties can be composed (section1.1.3), so that self-correspondences of X form a ring.

The following represents the starting point of Kimura’s work, and remainsopen.

Conjecture 2.2.7. (Nilpotence conjecture) Suppose that Γ ∈ CH(X × X) ishomologous to zero. Then there exists a positive integer N such that ΓN = 0in CH(X ×X)Q.

Kimura proves in [49] the remarkable result that this conjecture is equivalentto his finite dimensionality conjecture.

The following result is proved independently by Voevodsky [81] and Voisin[84].

Theorem 2.2.8. The nilpotence conjecture holds for cycles in X×X which arealgebraically equivalent to 0.

Proof. Let Γ ∈ CHd(X × X), d = dim X, be algebraically equivalent tozero. This means that there is a curve C that we may assume to be smooth, a 0-cycle z ∈ CH0(C) homologous to 0, and a correspondence Z ∈ CHd(C×X×X)such that

Γ = Z∗(z) in CHd(X ×X).


For any integer k, we can construct a correspondence Zk ∈ CHd(Ck ×X ×X)obtained using the composition of the cycles Zt, t ∈ C, namely we define Zk bythe formula

Zk(t1, . . . , tk) = Z(t1) . . . Z(tk), t1, . . . , tk ∈ C.

By definition, we getΓk = Zk∗(zk),

where the product zk ∈ CH0(Ck) is defined as pr∗1z · . . . · pr∗kz.The proof concludes with the following easy fact (see [86, II,Lemma 11.33]):

Lemma 2.2.9. For a zero cycle z homologous to 0 on a smooth curve C, thecycle zk vanishes in CH0(Ck) for k large enough.

The main concrete result concerning Conjecture 2.2.7 was established byKimura [49] who proved the following statement.

Theorem 2.2.10. (Kimura 2005) If X is dominated by a product of curvesthen X satisfies Conjecture 2.2.7.

The proof of this theorem is rather tricky, and would be out of place in here.If X is a surface, it is in fact sufficient that X be rationally dominated by theproduct of two curves to conclude the validity of Conjecture 2.2.7.

We prove next the following beautiful application:

Theorem 2.2.11. Conjecture 2.2.7 implies Bloch’s conjecture 0.2.2 for surfaceswith pg = q = 0. In particular, Bloch’s conjecture is valid for surfaces withpg = q = 0 that are rationally dominated by curves.

Proof. Let X be a surface satisfying these assumptions. By Lefchetz’stheorem on (1, 1)-classes, since pg(X) = 0, we know that

H2B(X, Z) = 〈[Ci]〉

is generated by classes of curves. Since q = 0, the Kunneth decomposition ofthe diagonal takes the following simple form:

[∆] ∈ H0B(X,Q)⊗H4

B(X,Q) ⊕ H2B(X,Q)⊗H2

B(X,Q)⊕H4B(X,Q)⊗H0

B(X,Q),

and as H2B(X,Q) is generated by the classes [Ci], we may write for any chosen

point x ∈ X

[∆X ] = [X × x] +∑

nij [Ci × Cj ] + [x ×X]. (2.2.17)

Consider the 2-cycle

Γ = ∆X − (X × x)−∑

nijCi × Cj − (x ×X) (2.2.18)


in X × X. Then by (2.2.17), [Γ] = 0. The nilpotence conjecture implies thatthere exists a positive integer N such that ΓN = 0 in CH2(X ×X)Q. Hence,

(Γ∗)N = (ΓN )∗ : CH0(X)Q → CH0(X)Q

is zero. But Γ∗ acts as the identity on the group CH0(X)hom of 0-cycles ofdegree 0, since ∆X acts as the identity, but the other terms in the right handside of (2.2.18) act trivially on CH0(X)hom. (For example, γ = X × x actstrivially on CH0(X)hom because γ∗(z) = (deg z)x.) Thus CH0(X)hom = 0.

Remark 2.2.12. Looking more closely at the proof and introducing Murre’sChow-Kunneth decomposition [63], one sees that the proof above would showas well that Conjecture 2.2.7 implies Bloch’s conjecture 0.2.4 for surfaces withpg = 0.

The next theorem is a direct generalization of Theorem 2.2.11. It proves thegeneralized Bloch conjecture 0.2.4 and Conjecture 2.2.4 for c = 1, under thegeneralized Hodge conjecture and the nilpotence conjecture:

Theorem 2.2.13. Let X be a smooth projective variety with hk,0(X) = 0 fork > r. Assume that:

(i) The generalized Hodge conjecture holds for X in coniveau 1.(ii) The Hodge conjecture is true for Y ×X with dim Y ≤ dim X.(iii) X satisfies the nilpotence conjecture 2.2.7.Then CH0(X) is supported on a closed algebraic subset Xr ⊂ X of dimension

≤ r.

Proof. (Sketch) Let us do it first for r = 0 (this is the situation of Conjecture2.2.4 with c = 1). We work again with the diagonal ∆X ⊂ X × X and itscohomology class

[∆X ] ∈ H0B(X, Q)⊗H2n

B (X, Q)⊕ · · ·

We have

[∆X ] = [X × x] mod⊕

k>0

HkB(X, Q)⊗H2n−k

B (X, Q).

The generalized Hodge conjecture implies that there exist Y of codimension 1and a resolution i : Y → Y → X, so that

i∗ : Hk−2B (Y ,Q) → Hk

B(X,Q).

is surjective for any k > 0, as follows from Theorem 1.2.20. Hence

[∆X −X × x] ∈ Im ((i, id)∗ : H2n−2B (Y ×X,Q) → H2n

B (X ×X,Q)).

On the right we have a Gysin map and thus a morphism of Hodge structures.


Corollary 1.2.5 implies that

[∆X −X × x] = (i, id)∗β, (2.2.19)

where β is a Hodge class on Y × X. We now finish the argument as before:the Hodge conjecture on Y ×X implies that β is the class [Z] of a cycle Z onY ×X. Put

Γ = ∆X −X × x − (i, id)∗Z,

so that [Γ] = 0 by (2.2.19). It follows that Γ∗ is nilpotent, yet Γ∗ acts asthe identity on CH0(X)hom, since Z∗ = 0 on CH0(X)Q (because Z ⊂ Y × Xwith Y X) and for z ∈ CH0(X)Q, (X × x)∗(z) = (deg z) x in CH0(X)Q.Therefore CH0(X)Q = Q and this implies that CH0(X) = Z because CH0(X)has no torsion under our assumptions by Roitman’s theorem (cf. [73]). Indeed,our assumptions imply that AlbX = 0.

For the general case, (r arbitrary), we just have to add the following argu-ment:

First of all, as we assumed that the Hodge conjecture holds for X ×X, theKunneth components of the diagonal are algebraic, so we can write:

[∆X ] =∑

i

δi in H2nB (X ×X,Q),

with δi = [Zi], Zi ∈ CHn(X ×X)Q and δi ∈ HiB(X,Q)⊗H2n−i

B (X,Q).For i > r, we know by assumption that the Hodge coniveau of Hi

B(X,Q) is atleast 1 and thus by the same arguments as above, we conclude using assumptionsi) and ii) that we may assume the Zi’s are supported on Y ×X where Y $ X isa proper closed algebraic subset. Assume next i ≤ r. Then 2n− i ≥ 2n− r, andthus by Lefschetz theorem on hyperplane sections, if we choose for Xr a smoothcomplete intersection of n−r ample hypersurfaces in X, and denote by jr : Xr →X the inclusion, the Gysin map jr∗ : Hk−2n+2r

B (Xr,Q) → HkB(X,Q) is surjective

for k ≥ 2n − r. Arguing as before, we conclude that under assumption ii), wemay assume the cycles Zi for i ≥ r are supported on X ×Xr. In conclusion, wefind assuming i) and ii) a decomposition of cycle classes:

[∆X ] = [Z1] + [Z2] in H2nB (X ×X,Q)

where Z1 ∈ CHn(X ×X)Q is supported on Y ×X and Z2 ∈ CHn(X ×X)Q issupported on X ×Xr.

We then conclude using assumption iii) that the cycle ∆X − Z1 − Z2 ∈CHn(X × X)Q, being cohomologous to 0, is nilpotent, and as Z1∗ = 0 onCH0(X)Q and Im Z2∗ : CH0(X)Q → CH0(X)Q is supported on Xr this impliesthat jr∗ : CH0(Xr)Q → CH0(X)Q is surjective.

Chapter 3

Chow groups of largeconiveau completeintersections

Our goal in this chapter, which is mainly based on [92] and [94], is to investigatethe generalized Bloch conjecture for hypersurfaces or complete intersections ineither projective space, or more generally varieties with trivial Chow groups. Wewill first recall following Griffiths [40] how the Hodge coniveau of such completeintersections is computed. We then turn to the study of coniveau 2 completeintersections in projective space, for which neither the generalized Hodge con-jecture nor the generalized Bloch conjecture are known to hold. We will sketch astrategy to attack the generalized Hodge conjecture for coniveau 2, which does notuse Theorem 2.2.3, hence does not pass through the computations of the Chowgroups CH0 and CH1. The main result proved in this chapter is Theorem 3.3.2,which says that, assuming Conjecture 1.2.10 (or Lefschetz standard conjecture),for a very general complete intersection Y of very ample hypersurfaces in a va-riety with trivial Chow groups, if the cohomology of Y has geometric coniveau≥ c, then its Chow groups CHi(Y )Q) are trivial for i ≤ c − 1. In particular,proving the generalized Hodge conjecture for such a Y is equivalent to provingthe generalized Bloch conjecture.

3.1 Hodge coniveau of complete intersections

Consider a smooth complete intersection X ⊂ Pn of r hypersurfaces of degreed1 ≤ . . . ≤ dr. By Lefschetz hyperplane section theorem (cf. [86, ??]), theonly interesting Hodge structure is the Hodge structure on Hn−r

B (X,Q), andin fact on the primitive part of it (that is the orthogonal of the restriction ofH∗

B(Pn,Q) with respect to the intersection pairing). We will say that X hasHodge coniveau c if the Hodge structure on Hn−r

B (X,Q)prim has coniveau c.

55

56 CHAPTER 3. COMPLETE INTERSECTIONS OF LARGE CONIVEAU

The Hodge coniveau of a complete intersection in projective space is com-puted as follows:

Theorem 3.1.1. X has coniveau ≥ c if and only if

n ≥r∑

i

di + (c− 1)dr. (3.1.1)

Conjecture 2.2.4 thus predicts the following statement for complete intersec-tions in projective space:

Conjecture 3.1.2. Let X be a complete intersection X ⊂ Pn of r hypersurfacesof degree d1 ≤ . . . ≤ dr. Then if n ≥ ∑

i di + (c − 1)dr, the cycle class map clis injective on CHi(X)Q for i ≤ c− 1.

The generalized Hodge conjecture 1.2.21 on the other hand predicts thefollowing:

Conjecture 3.1.3. Let X be a complete intersection X ⊂ Pn of r hypersurfacesof degree d1 ≤ . . . ≤ dr. Then if n ≥ ∑

i di +(c−1)dr, the primitive cohomologyHn−r

B (X,Q)prim is supported on a closed algebraic subset of codimension ≥ c.

In the next subsection, we will recall the proof of Theorem 3.1.1. The restof the chapter will be devoted to the study of Conjectures 3.1.2 and 3.1.3. Notethat according to Theorem 2.2.3, Conjecture 3.1.2 implies Conjecture 3.1.3.

Our main result in this chapter is taken from [94]. It says conversely that,assuming Conjecture 1.2.10, if a very general complete intersection X as abovesatisfies Conjecture 3.1.3, then it satisfies Conjecture 3.1.2.

Let us conclude this section by exhibiting for every d, following [85], smoothhypersurfaces X of degree d in Pn satisfying the conclusion of Conjecture 3.1.2.The examples constructed there are as follows: We write n = cd + s and choosehomogeneous degree d polynomials

f1 ∈ C[X0, . . . , Xd], f2, . . . , fc−1 ∈ C[Y1, . . . , Yd], fc ∈ C[Y1, . . . , Yd+s].

We set

f = f1(X0, . . . , Xd) + f2(Xd+1, . . . , X2d) + . . . + fc−1(X(c−2)d+1, . . . , X(c−1)d) (3.1.2)+fc(X(c−1)d+1, . . . Xcd+s).

The hypersurface X defined by f is smooth if and only if the hypersurfacesdefined by the fi’s are smooth.

Theorem 3.1.4. (Voisin 1996, [85]) Let X be as above and smooth. Then thecycle class map cl is injective on CHi(X)Q for i ≤ c− 1.

The proof which we will give below is closer to [34], [65]. It is in fact aconsequence of the following result:

3.1. HODGE CONIVEAU OF COMPLETE INTERSECTIONS 57

Theorem 3.1.5. [34] Let X be a smooth hypersurface of Pn, covered by a familyof projective spaces Pr. Then the groups CHl(X)hom are torsion for l ≤ r − 1,and CHr(X)Q is generated as a group by the classes of the Pr’s in the family.

Proof. The proof first uses the following observation.

Lemma 3.1.6. Let h = c1(OX(1)). Then the map

dh : CHl(X)hom → CHl−1(X)hom

z 7→ d z · h,

is equal to 0.

Proof. Indeed, by the definition of the map j∗, when j is the inclusion of aCartier divisor, and by the fact that the class of X in PicPn is equal to dh, wefind that

j∗ j∗ = dh : CHl(X)hom → CHl−1(X)hom.

But as CH∗(Pn)hom = 0, the map

j∗ : CHl(X)hom → CHl(Pn)hom

is zero.

Let now F be the considered family of r-planes contained in X. Up toreplacing F by a desingularization, we may assume that F is smooth.

Let Q = (x, P ) ∈ X × F | x ∈ P be the incidence variety, and p, q theprojections.

Qq //

p

²²

X

F.

By hypothesis, q is surjective, and up to replacing F by a subvariety, we mayassume that q is generically finite of degree N > 0. Then we know that theimage of the map

q∗ : CHl(Q)hom → CHl(X)hom

contains NCHl(X)hom, since q∗ q∗ = N · Id on CHl(X)hom.Moreover, the projection p makes Q into a Pr-bundle on F , and q∗(OX(1)) is

a divisorOQ(1) on this projective bundle Q → F . Writing H = c1(q∗(OX(1))) ∈CH1(Q), we can thus apply the computation of Chow groups of a projectivebundle (cf. [35, 3.3], [86, ??], which yields

CHl(Q) = ⊕0≤k≤r,l−r+k≥0Hkp∗CHl−r+k(F ).

By the fact that we also have the corresponding decomposition on the cohomol-ogy

HmB (Q,Z) = ⊕0≤k≤r,2k≤m[H]kp∗Hm−2k

B (F,Z), [H] ∈ H2B(Q,Z)


for every m (cf. [86, I,7.3.3]), we immediately deduce the decomposition

CHl(Q)hom = ⊕0≤k≤r,l−r+k≥0Hkp∗CHl−r+k(F )hom.

In particular, for l < r, the map

H : CHl+1(Q)hom → CHl(Q)hom

is surjective.Thus, for Z ∈ CHl(X)hom, with l < r, there exist Z ′ ∈ CHl(Q)hom, Z ′′ ∈

CHl+1(Q)hom such that

NZ = q∗Z ′ ∈ CHl(X), Z ′ = HZ ′′ ∈ CHl(Q). (3.1.3)

Now, if Z ′′ ∈ CHl+1(Q), then by the projection formula, we have

q∗(H · Z ′′) = q∗(q∗h · Z ′′) = h · q∗Z ′′ in CHl(X), (3.1.4)

where h := c1(OX(1)) ∈ CH1(X). Finally, we obtain from (3.1.3) and (3.1.4)

dNZ = d hq∗Z ′′

which is equal to 0 by lemma 3.1.6.For l = r, the above reasoning shows that the map

CH0(F )hom,Q ⊕ CHr+1(X)hom,Q → CHr(X)hom,Q

(z, Z) 7→ q∗p∗z + h · Zis surjective. Applying Lemma 3.1.6, we conclude that

CH0(F )hom,Q → CHr(X)hom,Q

is surjective, as desired.

Proof of Theorem 3.1.4. Using Theorem 3.1.5, the proof finishes observ-ing that an hypersurface X as above is covered by a family of linear spacesL ∼= Pc−1 which have the property that there is a linear space L′ ⊂ Pn withL′ ⊂ X = dL. Having this, we know on the one hand that the classes of theseL’s generate CHc−1(X)Q and on the other hand, their classes are all equal tohn−c

d in CHc−1(X)Q. Thus CHc−1(X)Q = Q and CHc−1(X)hom,Q = 0.Let us exhibit these spaces L′. We will do it for c = 2. Let P1

∼= Pd

and P2∼= Pd−1+s ⊂ Pn, n = 2d + s, be the linear subspaces of Pn defined by

Xi = 0, i > d (resp. Xi = 0, i ≤ d. Let x ∈ X, which we write as (x1, xc)according to the decomposition of the set of coordinates appearing in (3.1.2).We may assume that both xi’s are nonzero vectors. The hypersurface X1 ⊂ P1

defined by f1 is Fano, and it is a result due to Roitman [74] that for any x1 ∈ X1

there exists a line ∆ in Pd passing through x1 and such that f1|∆ = Ud, whereU is a linear equation defining x1 in ∆. The line ∆ contains the point x1. Let


P ′2 := 〈x1, P2〉 ∼= Pd+s ⊂ Pn. The point x belongs to P ′2 and again by Roitman’sresult, there is a line ∆′ ⊂ P ′2 passing through x such that f|P ′2 = V d. LetP := 〈∆, ∆′〉. Then P ∼= P3, x ∈ P and the restriction of f to P is of the formW d − Ud, for linear forms W, U on P vanishing at x. Restricting to the planP ′ ⊂ P defined by W − U = 0, we find that f|P ′ = Zd for some linear form onP ′ vanishing at x.

3.1.1 Proof of Theorem 3.1.1 in the case of hypersurfaces

We prove Theorem 3.1.1 using Griffiths method comparing the pole order andHodge filtration (see [40] or [86], II, 6.1.2). Let W be a projective variety, and

Xl

→ W a smooth hypersurface. In the following, all varieties are endowedwith the classical topology and analytic structural sheaves, so the notation ΩW

below is what we denoted ΩWanin Section 1.2. Set

U = W −Xj

→ W.

Recall (cf. [86, I,8.2.2]) that the logarithmic de Rham complex (Ω•W (log X), d)is the complex of free OX−modules defined as follows:

ΩkW (log X) = ∧kΩX(log X),

where ΩW (log X) is the sheaf of free OW−modules generated locally by ΩW

and by df/f , where f is a local holomorphic equation for X. The differentialis the exterior differential. This complex can be viewed as a subcomplex of thecomplex j∗A•U , and we have the following result (see [86, I,8.2.3]).

Theorem 3.1.7. The inclusion

Ω•W (log X) → j∗A•U ,

where AkU is the sheaf of C∞ complex differential k-forms on U , is a quasi-

isomorphism.

We have two morphisms of complexes: the inclusion

Ω•W → Ω•W (log X),

which in cohomology induces the restriction

HkB(W,C) → Hk

B(U,C), (3.1.5)

and the residueRes : Ω•W (log X) → Ω•−1

X ,

which to α∧ (df/f) associates Res(α∧ dff ) = 2iπα|X . The map induced by Res

in cohomology is the topological residue

HkB(U,C) → Hk−1

B (X,C), (3.1.6)


which we can also define as the composition

HkB(U,C) → Hk+1

B (W,U,C) ∼= Hk+1(T, ∂T ) ∼= Hk−1B (X,C),

where T is a tubular neighbourhood of X in W , the first arrow is the connectionmorphism in the long exact sequence of relative cohomology, the second is theexcision isomorphism, and the last is the Thom isomorphism.

The morphisms (3.1.5) and (3.1.6) are defined on Z and compatible with theHodge filtrations (cf. Section 1.2), because the Hodge filtration on H∗

B(U,C),resp. H∗

B(X,C) is induced by the naıve filtration on the complex Ω•W (log X),resp. Ω•X . (More precisely, the morphism Res sends F pHk

B(U,C) to F p−1Hk−1B (X,C).)

Let us now assume that X ⊂ W is ample and that the pair satisfies thefollowing conditions:

For every k > 0, i > 0, j ≥ 0, we have (3.1.7)Hi(W,Ωj

W (kX)) = 0.

These hypotheses are satisfied by Bott’s vanishing theorem if W is the pro-jective space Pn. Under these hypotheses, we have the following result.

Theorem 3.1.8. (Griffiths [40]) For every integer p such that 1 ≤ p ≤ n =dim W , the image of the natural map

H0(W,KW (pX)) → HnB(U,C) (3.1.8)

which to a section α (viewed as a meromorphic form on W of degree n, andtherefore closed, holomorphic on U and having a pole of order p along X) asso-ciates its de Rham cohomology class, is equal to Fn−p+1Hn

B(U,C).

We refer to [40], [86, ??] for the proof of this result.Now assume that W is the projective space Pn, and that X is a smooth

hypersurface of degree d, of equation f = 0. We know that KPn = OPn(−n−1),a generator of H0(Pn,KPn(n + 1)) being given by

Ω =∑

i

(−1)iXidX0 ∧ . . . ˆdXi . . . ∧ dXn

= X0 . . . Xn

∑

i

(−1)i dX0

X0∧ . . .

ˆdXi

Xi. . . ∧ dXn

Xn,

where the Xi’s are homogeneous coordinates on Pn. As OPn(X) = OPn(d),Theorem 3.1.8 shows that for every p such that 1 ≤ p ≤ n, we have a surjectivemap

αp : H0(Pn,OPn(pd− n− 1)) → Fn−p+1HnB(U,C)

∼= Fn−pHn−1B (X,C)prim,

which to a polynomial P associates the residue of the class of the meromorphicform PΩ

fp .


In particular, if n ≥ dc, we find that H0(Pn,OPn(pd − n − 1)) = 0 forp ≤ c, hence that Fn−cHn−1

B (X,C)prim = 0. This proves the “if” part inTheorem 3.1.1 for hypersurfaces, because the vanishing of Fn−cHn−1

B (X,C)prim

is equivalent to the fact that the coniveau of the weight n− 1 Hodge structureon Hn−1

B (X,C)prim is ≥ c. We refer to [86, ??] for the “only if” part.

3.1.2 Complete intersections

In order to get Theorem 3.1.1 for complete intersections, we reduce to the caseof hypersurfaces by the following trick (cf. [79], [32]).

Let L1, . . . , Lr be ample hypersurfaces on W , with W smooth projective ofdimension n. Let E := L1 ⊕ . . . ⊕ Lr and WE := P(E), which is equipped withthe line bundle

OWE(1) := OP(E)(1)

satisfying the property that π∗OWE(1) = E where π : WE → W is the structural

map (so we follow the Grothendieck convention here). Given hypersurfacesXi ∈ |Li| defined by equations fi ∈ H0(W,Li), we get a section τ := (f1, . . . , fr)of E . Identifying H0(W, E) to H0(WE ,OWE

(1)), we thus get an hypersurface

XE ⊂ WE , XE ∈ |OWE (1)|,

defined by an equation f ∈ H0(WE ,OWE(1)). The hypersurface XE is ample

since the vector bundle E is ample on W . Furthermore XE is smooth if andonly if X := ∩iXi is a smooth complete intersection.

The canonical bundle of WE is computed by the formula (which is a conse-quence of the relative Euler exact sequence):

KWE = OWE (−r) + π∗det E + π∗KW .

We have det E = ⊗iLi.When W = Pn, which we assume from now on, the line bundles Li are of

the form O(di), di > 0 and the vanishing conditions (3.1.7) are satisfied by theline bundle OWE

(1), as a consequence of Bott’s vanishing theorem on Pn.Let Ω be a trivializing section of the line bundle KWE

⊗OWE(r)⊗π∗O(−∑

i di+n + 1). Theorem 3.1.8 shows that for every p such that 1 ≤ p ≤ n + r − 1, wehave a surjective map

αp : H0(WE ,OWE(p− r)⊗ π∗O(+

∑

i

di − n− 1)) → Fn+r−pHn+r−1B (U,C)

∼= Fn+r−2Hn+r−1B (XE ,C)van,

which to a section P associates the residue of the class of the meromorphic formPΩfp .

We thus deduce that for the hypersurface XE we have

Fn+r−p−1Hn+r−2B (XE ,C)van = 0 (3.1.9)


if H0(WE ,OWE(p− r)⊗ π∗O(

∑i di − n− 1)) = 0.

We now compute

H0(WE ,OWE(p− r)⊗ π∗O(

∑

i

di − n− 1)) (3.1.10)

= H0(Pn, (Sp−r(⊕iOPn(di)))⊗O(∑

i

di − n− 1)),

where Si denotes the i-th symmetric power of the considered vector bundle.Combining (3.1.9) and (3.1.10) we conclude that

Fn+r−p−1Hn+r−2B (XE ,C)van = 0 if

∑

i

di + (p− r)Sup di − n− 1 < 0.

To conclude the proof of Theorem 3.1.1 (or rather of the “if” part) forcomplete intersections, it just remains to show the following

Proposition 3.1.9. [32], [79] The hypersurface XE ⊂ WE contains the Pr−1-bundle P(E|X). Let

j : P(E|X) → XE , π : P(E|X) → X

be the natural maps. Then the morphism

j∗ π∗ : Hn−rB (X,Q)prim → Hn+r−2

B (X,Q)van

is an isomorphism of Hodge structures of bidegree (r − 1, r − 1).

Proof. The morphism above is obviously a morphism of Hodge struc-tures. It suffices thus to prove that it induces an isomorphism between thesegroups. It is not hard to see that it sends the primitive part Hn−r

B (X,Q)prim

to the vanishing part Hn+r−2B (XE ,Q)van. To see that it induces an isomor-

phism between both, we introduce the open set V := Pn \X and observe thatπ : U = WE \XE → V = Pn \X is an affine bundle (with fiber Pr−1 \ fw = 0over w ∈ W = Pn). It follows that π∗ induces an isomorphism

π∗ : HnB(V,Q) ∼= Hn

B(U,Q).

The result is then obtained by comparing the relative exact sequences of thepairs (WE , U) and (W,V ): We have the following commutative diagram of exactsequences

H∗B(W,Z) //

²²

H∗B(V,Z) //

'²²

H∗−2r+1B (X,Z)

²²

j∗ // H∗+1B (W,Z) . . .

²²H∗

B(WE ,Z) // H∗B(U,Z) // H∗−1

B (XE ,Z)jE∗ // H∗+1

B (WE ,Z) . . .

(3.1.11)

Here we identified by Thom’s isomorphism the relative cohomology H∗+1B (WE , U,Z)

to H∗−1B (XE ,Z) and the relative cohomology H∗+1

B (W,V,Z) to H∗−2r+1B (X,Z).


The vertical maps are all pull-back maps π∗. The map jE denotes the inclusionof the hypersurface XE in WE .

As we already noticed, the second pull-back map π∗ is an isomorphism inall degrees. Furthermore, by definition of vanishing cohomology, the kernels ofthe maps j∗, jE∗ are H∗−2r+1

B (X,Z)van, H∗−2r+1B (XE ,Z)van respectively. Thus

our diagram becomes the following exact diagram :

0 // H∗B(W,Z)/Im j∗ //

²²

H∗B(V,Z) //

'²²

H∗−2r+1B (X,Z)van

²²

// 0

0 // H∗B(WE ,Z)/Im jE∗ // H∗

B(U,Z) // H∗−1B (XE ,Z)van

// 0

(3.1.12)

We now consider what happens for ∗ = n + r − 1. Then we claim that the twogroups Hn+r−1

B (W,Z)/Im j∗ and Hn+r−1B (WE ,Z)/Im jE∗ are naturally isomor-

phic via the map π∗. Indeed, we recall that XE is an hypersurface in the amplelinear system |OWE

(1)|. Using Lefschetz theorem on hyperplane sections (cf.[86, ??]), it follows that

Im jE∗ : Hn+r−3B (XE ,Z) → Hn+r−1

B (WE ,Z)

is equal toIm l∪ : Hn+r−3

B (WE ,Z) → Hn+r−1B (WE ,Z),

where l := c1(OWE(1)). We use now the fact that WE is a Pr−1-bundle over W ,

so that its cohomology is described as follows:

H∗B(WE ,Z) = ⊕i≤r−1l

i ∪ π∗H∗−2iB (W,Z)

so thatH∗

B(WE ,Z)/Im l∪ ∼= H∗B(W,Z)/Im cr∪,

where c is the constant term of the Chern polynomial satisfied by l:

lr = −∑

i≤r−1

(−1)r−ili ∪ π∗cr−i(E).

On the other hand, the class of X in W is equal to cr(E). It thus follows that

Im cr∪ : Hn−r−1B (W,Z) → Hn+r−1

B (W,Z) (3.1.13)

is contained in

Im j∗ : Hn−r−1B (X,Z) → Hn+r−1

B (W,Z). (3.1.14)

If we now recall that E is a direct sum of ample line bundles on W , we have thatX is a complete intersection of ample hypersurfaces in W , hence the restrictionmap

j∗ : Hn−r−1B (W,Z) → Hn−r−1

B (X,Z)


is surjective by Lefschetz theorem on hyperplane sections and (3.1.14) is in factequal to (3.1.13), which concludes the proof of the claim.

We then conclude from diagram (3.1.12) and the last claim that

Hn−rB (X,Z)van

∼= Hn+r−2B (XE ,Z)van.

3.1.3 Coniveau 1 and further examples

For c = 1, the estimate in Theorem 3.1.1 is obvious, as coniveau(X) ≥ 1 isequivalent to Hn−r,0(X) = H0(X,KX) = 0, that is, X is a Fano completeintersection. In this case, the generalized Hodge-Grothendieck conjecture 1.2.21for the coniveau 1 Hodge structure on Hn−r(X,Q)prim is known to be true. Letus give two proofs of this.

First of all, we can do it explicitly using the correspondence between X andits Fano variety of lines F . Denoting by

Pq→ X

p ↓F

the incidence correspondence, where p is the tautological P1-bundle on F , onecan show (see for example [77]) that taking a n − r − 2-dimensional completeintersection Fn−r−2 ⊂ F and restricting P to it, the resulting morphism ofHodge structures

q′∗ p′∗ : Hn−r−2B (Fn−r−2,Q) → Hn−r

B (X,Q)

is surjective, where P ′ = p−1(Fn−r−2) and p′, q′ are the restrictions of p, q toP ′. It follows that Hn−r

B (X,Q) vanishes on the complement of the (singular)hypersurface q(P ′) ⊂ X.

The second proof is by a direct application of Theorem 2.1.12. Indeed,Fano varieties are rationally connected (cf. [55]) hence have trivial CH0 group.Theorem 2.1.12 then says that their cohomology of positive degree is supportedon a proper closed algebraic subset.

3.2 Coniveau 2 complete intersections

3.2.1 A conjecture on effective cones

Let Y be smooth projective, and denote, for any integer k, H2kB (Y,R)alg the sub-

space of H2kB (Y,R) generated over R by cycle classes and E2k(Y ) ⊂ H2k

B (Y,R)alg

the effective cone, that is the convex cone generated by classes of subvarietiesZ ⊂ Y of codimension k.

Cones of effective cycles have been very much studied in codimension 1or in dimension 1 (cf [15]), but essentially nothing is known in intermediate

3.2. CONIVEAU 2 COMPLETE INTERSECTIONS 65

(co)dimensions. Let us say that an algebraic cohomology class is big if it belongsto the interior of the effective cone. Note the following easy lemma (cf. [92]):

Lemma 3.2.1. Let h ∈ H2(Y,R) be the first Chern class of an ample linebundle on Y . A class α ∈ H2k

B (Y,R)alg is big if and only if, for some ε > 0,α− εhk ∈ E2k(Y ).

Proof. Indeed, the “only if” is obvious, because if α is in the interiorof E2k(Y ), a small deformation α − εhk also belongs to E2k(Y ). In the otherdirection, it suffices to prove that hk belongs to the interior of the effective cone.This is true because for any variety Z ⊂ Y of codimension k, there is an integerNZ such that Nhk−Z is effective. Thus both hk− 1

NZ[Z] and hk+ 1

NZ[Z] belong

to the cone E2k(Y ). As the classes [Z] generate the vector space H2kB (Y,R)alg,

we can choose a basis [Zi] of H2kB (Y,R)alg, and letting ε = Inf 1

NZi, we get

that the open neighborhood

hk +∑

i

εi[Zi], |εi| < ε

of hk in H2kB (Y,R)alg is contained E2k(Y ). The proof is finished.

In [68], it is shown that when dimW = 1, and W ⊂ V is moving and hasample normal bundle, its class [W ] is big. We give in [92] an example workingin any dimension ≥ 4 and in codimension 2, showing that in higher dimensions,a moving variety W ⊂ V with ample normal bundle may not have a big class.Here by moving, we mean that a generic deformation of W in V may be imposedto pass through a generic point of V .

Definition 3.2.2. A smooth k-dimensional subvariety V ⊂ Y , is very movingif it has the following property: through a general point y ∈ Y , and given ageneral vector subspace W ⊂ TY,y of rank k, there is a deformation V ′ ⊂ Y ofV in Y which is smooth and passes through y with tangent space equal to W aty.

We make the following conjecture for “very moving” subvarieties.

Conjecture 3.2.3. Let Y be smooth and projective and let V ⊂ Y be a verymoving subvariety. Then the class [V ] of V in Y is big.

3.2.2 On the generalized Hodge conjecture for coniveau 2complete intersections

Let X ⊂ Pn be a generic complete intersection of multidegree d1 ≤ . . . ≤ dr.So X has dimension n− r and the interesting cohomology of X is supported indegree n− r. Let F be the variety of lines contained in X. Then by genericityF is smooth of dimension 2n − 2 − ∑

i di − r. Note that if n ≥ ∑i di, the

morphism P∗ : Hn−r−2B (F,Q) → Hn−r

B (X,Q) induced by the universal P1-bundle P ⊂ F ×X is surjective (cf. Section 3.1.3).


For a generic section G ∈ H0(X,OX(n−∑i di− 1)) with zero set XG ⊂ X,

consider the subvariety FG ⊂ F of the variety of lines contained in XG. Again bygenericity, FG is smooth of dimension n−r−2. The following result is obtainedin [92] by studying the deformations of FG in F induced by deformations of G.

Proposition 3.2.4. When n ≥ ∑i di + dr, that is when X has Hodge coniveau

≥ 2, the subvariety FG ⊂ F is very moving in the sense of Definition 3.2.2.

Let us now, following [92], deduce from this proposition the following result:

Theorem 3.2.5. (Voisin 2010) If the very moving varieties FG satisfy conjec-ture 3.2.3, that is [FG] is big, then the generalized Hodge conjecture for coniveau2 is satisfied by X.

Proof. Assume that [FG] is big. Then it follows by lemma 3.2.1 that forsome positive large integer N and for some effective cycle E of codimensionn−∑

i di on F , one has:

N [FG] = ln−∑

i di + [E], (3.2.15)

where l is the restriction to F of first Chern class of the Plucker line bundle onthe Grassmannian G(1, n). (Here we could work as well with real coefficients,but as we want actually to do geometry on E, it is better if E is a true cycle.) Fora ∈ Hn−r

B (X,Q)prim, let η = p∗q∗a ∈ Hn−r−2B (F,Q) We use the fact, essentially

due to Shimada [77], (see also [92, Lemma 1.1]) that η is primitive with respectto l and furthermore vanishes on FG, with dimFG = n− r − 2.

Let us now assume that a ∈ Hp,q(X)prim and integrate (−1)k(k−1)

2 ip−qη ∪η, k = p + q − 2 = n− r − 2 over both sides in (3.2.15). We thus get

0 =∫

F

(−1)k(k−1)

2 ip−qln−∑

i di ∪ η ∪ η +∫

E

(−1)k(k−1)

2 ip−qη ∪ η.

As η is primitive with respect to l, and non zero if a is non zero, by the secondHodge-Riemann bilinear relations (cf [86], I, 6.3.2), we have

∫F(−1)

k(k−1)2 ip−qln−

∑i di∪

η ∪ η > 0. It thus follows that∫

E

(−1)k(k−1)

2 ip−qη ∪ η < 0. (3.2.16)

Let E = tEj be a desingularization of the support of E =∑

j mjEj , mj > 0.Thus we have ∑

j

mj

∫

Ej

(−1)k(k−1)

2 ip−qη ∪ η < 0.

It thus follows from (3.2.16) that there exists at least one Ej such that∫

Ej

(−1)k(k−1)

2 ip−qη ∪ η < 0. (3.2.17)

3.3. ON THE GENERALIZED BLOCH AND HODGE CONJECTURES 67

Choose an ample divisor Hj on each Ej . By the second Hodge-Riemann bilinearrelations, inequality (3.2.17) implies that η|Ej

is not primitive with respect tothe polarization given by Hj , that is η ∪ [Hj ] 6= 0 and in particular

η|Hj6= 0.

In conclusion, we proved that the composed map

Hn−rB (X,Q)prim

p∗q∗→ Hn−r−2B (F,Q) →

⊕Hn−r−2

B (Hj ,Q)

is injective, where the second map is given by pull-back to Hj via the composedmap Hj → Ej → F . If we dualize this, recalling that dimHj = n − r − 3, weconclude that ⊕

Hn−r−4B (Hj ,Q) → Hn−r

B (X,Q)prim

is surjective, where we consider the pull-backs of the incidence diagrams to Hj

Pjqj→ X

pj ↓Hj

and the map is the sum of the maps qj∗p∗j , followed by orthogonal projectiononto primitive cohomology. As for n − r even and n − r ≥ 3, the image of∑

j qj∗p∗j also contains the class hn−r

2 , (up to adding if necessary to the Hj theclass of a linear section of F ) it follows immediately that the map

∑

j

qj∗p∗j :⊕

Hn−r−4B (Hj ,Q) → Hn−r

B (X,Q)

is also surjective, if n− r ≥ 3 (the case n− r = 2 is trivial). This implies thatHn−r

B (X,Q) is supported on the n− r− 2-dimensional variety ∪jqj(Pj), that isvanishes on X \ ∪jqj(Pj). The result is proved.

3.3 Equivalence of generalized Bloch and Hodgeconjectures for general complete intersec-tions

Our main result in this section concerns very general complete intersections Xb

in a variety X with trivial Chow groups, with the following meaning:

Definition 3.3.1. A smooth complex algebraic variety (non necessarily projec-tive) is said to have trivial Chow groups if it satisfies the property that the cycleclass map:

cl : CH∗(X)Q → H2∗(X,Q)

is injective.


Smooth projective varieties with trivial Chow groups include all smoothtoric varieties, eg the projective space, and varieties admitting a stratificationby affine spaces, like the Grassmannians.

We will prove for such Xb’s the generalized Bloch conjecture as formulatedin Remark 2.2.5. The statement is that if the transcendental cohomology ofa smooth projective variety Y has geometric coniveau c, then the cycle classmap is injective on cycles of Y of dimension ≤ c − 1. The generalized Blochconjecture 2.2.4 states that if the cohomology of Y has Hodge coniveau c moduloclasses of algebraic cycles, then the cycle class map is injective on cycles of Yof dimension ≤ c− 1. Thus the two statements differ by the generalized Hodgeconjecture 1.2.21.

We will prove in subsection 3.3.4 the following result:

Theorem 3.3.2. Assume the “standard” conjecture 1.2.10 holds for degree2n− 2r cycle classes. Let X be a smooth complex projective variety with trivialChow groups. Let L1, . . . , Lr, r ≤ n := dimX, be very ample line bundles on X.Assume that for a very general complete intersection Xb = X1 ∩ . . .∩Xr of hy-persurfaces Xi ∈ |Li|, the Hodge structure on Hn−r(Xb,Q)prim is supported ona closed algebraic subset Yb ⊂ Xb of codimension ≥ c. Then for the general suchXb (hence in fact for all), the cycle map cl : CHi(Xb)Q → H2n−2r−2i(Xb,Q) isinjective for any i < c.

There are two cases where the Conjecture 1.2.10 will be automatically sat-isfied, namely when the fibers Xb are surfaces or threefolds. Furthermore, inthe surface case, due to the Lefschetz (1, 1)-theorem, the geometric coniveau isequal to the Hodge coniveau. In the surface case, Theorem 3.3.2 proves Blochconjecture 2.2.4 for complete intersection surfaces inside a variety with trivialChow groups and also a variant of it for complete intersection surfaces withgroup action (cf. Section 3.3.5).

3.3.1 Varieties with trivial Chow groups

We start stating mostly without proofs a number of easy results concerningsmooth quasiprojective complex varieties X with trivial Chow groups. Themissing proofs can be found in [94].

Lemma 3.3.3. Let X be a smooth complex variety with trivial Chow groups.Then any projective bundle p : P(E) → X, where E is a locally free sheaf on X,has trivial Chow groups.

Lemma 3.3.4. Let X be a smooth complex algebraic variety with trivial Chowgroups and let Y ⊂ X be a smooth closed subvariety with trivial Chow groups.Then the blow-up XY → X of X along Y has trivial Chow groups.

Lemma 3.3.5. Assume Conjecture 1.2.10. Let X be a smooth projective varietywith trivial Chow groups. Then any Zariski open set U ⊂ X has trivial Chowgroups.


Proof. Write U = X \ Y . Let Z be a codimension k cycle on U withvanishing cohomology class. Then Z is the restriction to U of a cycle Z on X,which has the property that

[Z]|U = 0 in H∗B(U,Q).

Conjecture 1.2.10 says that there is a cycle Z ′ supported on Y such that [Z] =[Z ′] in H2k(X,Q). The cycle Z−Z ′ is thus cohomologous to 0 on X. As X hastrivial Chow groups, Z − Z ′ is rationally equivalent to 0 on X modulo torsion,and so is its restriction to U , which is equal to Z.

Let us conclude with two more properties:

Lemma 3.3.6. Let φ : X → X ′ be a projective surjective morphism, where Xand X ′ are smooth complex algebraic varieties. If X has trivial Chow groups,so does X ′.

Proposition 3.3.7. Let X be a smooth projective variety with trivial Chowgroups. Then X ×X has trivial Chow groups.

Proof. This uses the fact (proved in Section 2.2.1) that a variety with trivialChow groups has a complete decomposition of the diagonal as a combination ofproducts of algebraic cycles (cf. [56], [67], [86, II,10.3.1]):

∆X =∑

i,j

nijZi × Zj in CHn(X ×X)Q,

where nij ∈ Q, and dim Zi + dim Zj = n = dim X when nij 6= 0. It followsthat the variety Z := X × X also admits such decomposition, since ∆Z =p∗13∆X · p∗24∆X in CH2n(Z × Z), where pij is the projection of Z × Z = X4 tothe product X ×X of the i-th and j-th summand.

But this in turn implies that the cycle class map cl : CH∗(Z)Q → H2∗(Z,Q)is injective. Indeed, write

∆Z =∑

i,j

mijWi ×Wj in CH2n(Z × Z).

Then any cycle γ ∈ CH(Z)Q satisfies

γ = ∆Z∗γ =∑

i,j

mijdeg (γ ·Wi)Wj in CH(Z)Q.

It immediately follows that if γ is homologous to 0, it vanishes in CH(Z)Q.

The same proof shows as well the following result, as noticed by Lie Fu:

Proposition 3.3.8. Let X, Y be a smooth projective varieties with trivial Chowgroups. Then X × Y has trivial Chow groups.


3.3.2 A consequence of Conjecture 1.2.10

We first observe in this section that assuming the Lefschetz conjecture, or moreprecisely Conjecture 1.2.10, the generalized Hodge conjecture for sub-Hodgestructures on X can be rephrased by saying that a certain class on X × X issupported on a product Y × Y . This will be quite important for the rest of theproof.

Let X be a smooth complex projective variety of dimension n, and let Lbe a sub-Hodge structure of Hn

B(X,Q)prim, where the subscript “prim” standsfor “primitive with respect to a given polarization on X”. We know then bythe second Hodge-Riemann bilinear relations [86, I, 6.3.2] that the intersectionform < , > restricted to L is nondegenerate. Let πL : Hn

B(X,Q) → L be theorthogonal projector onto L. We assume that πL is algebraic, that is, there is an-cycle ∆L ⊂ X ×X, such that

[∆L]∗ = πL : HnB(X,Q) → L ⊂ Hn

B(X,Q),

[∆L]∗ = 0 : HiB(X,Q) → Hi

B(X,Q), i 6= n.

Lemma 3.3.9. Assume that there exists a closed algebraic subset Y ⊂ X suchthat L vanishes under the restriction map Hn

B(X,Q) → HnB(X \ Y,Q). Then if

Conjecture 1.2.10 holds, there is a cycle Z ′L ⊂ Y × Y with Q-coefficients suchthat

[Z ′L] = [∆L] in H2nB (X ×X,Q).

Proof. Indeed, because πL is the orthogonal projector on L, the class[∆L] belongs to L ⊗ L ⊂ H2n

B (X × X,Q). As L vanishes in HnB(X \ Y,Q),

the class [∆L] ∈ L ⊗ L vanishes in H2nB (X × X \ (Y × Y ),Q). Conjecture

1.2.10 then guarantees the existence of a cycle Z ′L ⊂ Y × Y such that [Z ′L] =[∆L] in H2n

B (X ×X,Q).

3.3.3 A spreading result

Let π : X → B be a smooth projective morphism and let (π, π) : X ×B X → Bbe the fibered self-product of X over B. Let Z ⊂ X ×B X be a codimension kalgebraic cycle. We denote the fibers Xb := π−1(b), Zb := Z|Xb×Xb

.

Proposition 3.3.10. Assume that for a very general point b ∈ B, there exista closed algebraic subset Yb ⊂ Xb ×Xb of codimension c, and an algebraic cycleZ ′b ⊂ Yb × Yb with Q-coefficients, such that


Then there exist a closed algebraic subset Y ⊂ X of codimension c, and a codi-mension k algebraic cycle Z ′ with Q-coefficients on X ×B X , which is supportedon Y ×B Y and such that for any b ∈ B,



Remark 3.3.11. This proposition is a crucial observation of the paper [94].The key point is the fact that we do not need to make any base change forthis specific problem. This will be crucial because the total space of the familyX ×B X is very easy to describe, while it can become very complicated afteran arbitrary base change. The idea of spreading-out cycles has become veryimportant in the theory of algebraic cycles since Nori’s paper [64], (cf. [39],[75]). For most problems however, we usually need to work over a genericallyfinite extension of the base, due to the fact that cycles existing at the generalpoint will exist on the total space of the family only after a base change.

Proof of Proposition 3.3.10. There are countably many algebraic vari-eties Mi → B parameterizing data (b, Yb, Z

′b) as above, and we can assume that

each Mi parameterizes universal objects

Yi → Mi, Yi ⊂ XMi, Z ′i ⊂ Yi ×Mi

Yi, (3.3.18)

satisfying the property that for m ∈ Mi, with pr1(m) = b ∈ B,

[Z ′i,b] = [Zi,b] in H2kB (Xb ×Xb,Q).

By assumption, a very general point of B belongs to the union of the imagesof the first projections Mi → B. By a Baire category argument, we concludethat one of the morphisms Mi → B is dominating. Taking a subvariety of Mi ifnecessary, we may assume that φi : Mi → B is generically finite. We may alsoassume that it is proper and carries the families Yi → Mi, Yi ⊂ XMi , Z ′i ⊂Yi ×Mi Yi. Denote by ri : XMi → X the proper generically finite morphisminduced by φi. Let

Y := ri(Yi) ⊂ X .

Note that because ri is generically finite, codimY ≥ c. Let r′i : Yi → Y be therestriction of ri to Yi. and let Z ′ := (r′i, r

′i)∗(Z ′i), which is a codimension k cycle

in X ×B X supported in

(r′i, r′i)(Yi ×Mi Yi) ⊂ Y ×B Y.

It is obvious that for any b ∈ B, [Z ′b] = N [Zb] in H2kB (Xb ×Xb,Q), where N

is the degree of ri.

3.3.4 Proof of theorem 3.3.2

We will start with a few preparatory Lemmas. Consider a smooth projectivevariety X of dimension n with trivial Chow groups. Let Li, i = 1, . . . , r, be veryample line bundles on X. Let j : Xb → X be a very general complete inter-section of hypersurfaces in |Li|, i = 1, . . . , r. Then Xb is smooth of dimensionn− r, and the Lefschetz theorem on hyperplane sections implies that

H∗B(Xb,Q) = Hn−r

B (Xb,Q)van ⊕H∗B(X,Q)|Xb

, (3.3.19)


where the vanishing cohomology Lb := Hn−r(Xb,Q)van is defined as Ker (j∗ :Hn−r

B (Xb,Q) → Hn+rB (X,Q)) and the direct sum above is orthogonal with

respect to Poincare duality on Xb. Indeed, for k < n − r the restriction mapHk

B(X,Q) → HkB(Xb,Q) is surjective, and choosing an ample line bundle L

on X with first Chern class l, the hard Lefschetz theorem on Xb says that∪ls|Xb

: Hn−r−sB (Xb,Q) → Hn−r+s

B (Xb,Q) is surjective for s ≥ 0. We thusdeduce that the restriction map Hk

B(X,Q) → HkB(Xb,Q) is also surjective for

k > n− r. Note that

Hn−rB (Xb,Q)van ⊂ Hn−r

B (Xb,Q)prim,

where “prim” means primitive for a very ample line bundle coming from X,and thus, by the second Hodge-Riemann bilinear relations, the intersection form< , > on Hn−r

B (Xb,Q) remains nondegenerate after restriction to Lb.As X has trivial Chow groups, we know that H∗

B(X,Q) is generated byclasses of algebraic cycles and so is the restriction H∗

B(X,Q)|Xb(cf. Theorem

2.2.2). This implies the following:

Lemma 3.3.12. The orthogonal projector πLbon Lb is algebraic.

Proof. In fact, we can construct an almost canonical algebraic cycle ∆b,van

with Q-coefficients on Xb × Xb whose class [∆b,van] is equal to πLb. For this,

we choose a basis of ⊕i≤n−rH2iB (X,Q). As we know that X has trivial Chow

groups, this basis consists of classes [zi,j ] of algebraic cycles zi,j on X, withcodim zi,j = i ≤ n− r. We choose an ample line bundle H on X, and note thatby the hard Lefschetz theorem applied to Xb, the classes [h]n−r−i ∪ [zi,j ]|Xb

,together with the classes [zi,j ]|Xb

, form a basis of H∗B(X,Q)|Xb

. Here [h] ∈H2

B(Xb,Q) is the topological first Chern class of H|Xb. The intersection form

on H∗B(Xb,Q) is nondegenerate when restricted to H∗

B(X,Q)|Xb, and Lb is the

orthogonal complement of H∗B(X,Q)|Xb

with respect to the intersection pairingon H∗

B(Xb,Q). We thus have the equality of orthogonal projectors:

πLb+ πH∗

B(X,Q)|Xb= IdH∗

B(Xb,Q).

But it is now clear that the orthogonal projector πH∗B(X,Q)|Xb

is given by theclass of an algebraic cycle on Xb×Xb, which is in fact almost canonical. Indeed,it suffices to choose an orthogonal basis of H∗

B(X,Q)|Xbfor the intersection form

on H∗B(Xb,Q) and to recall that H∗

B(X,Q)|Xbconsists of algebraic classes on

Xb. As IdH∗B(Xb,Q) corresponds to the class of the diagonal of Xb, the proof is

finished.

We now assume that there is a closed algebraic subset Yb ⊂ Xb of codi-mension c such that the sub-Hodge structure Lb vanishes on Xb \ Yb. Then,under Conjecture 1.2.10, Lemma 3.3.9 tells that there is an algebraic cycle Zb

supported on Yb×Yb such that [Zb] = [∆b,van]. We now put this information infamily.


Notation 3.3.13. Let X be a smooth projective with trivial Chow groups.Let Pi := P(H0(X, Li)). Let B ⊂ ∏

i Pi be the open set parameterizing smoothcomplete intersections and let

X ⊂ B ×X, π : X → B,

be the universal family. We will denote Xb ⊂ X the fiber π−1(b) for b ∈ B.

We will apply Proposition 3.3.10 to Z = Dvan, the corrected relative diagonalwith fiber over b ∈ B the ∆b,van introduced in Lemma 3.3.12. (Note that Dvan

is not in fact canonically defined even if its restriction to Xb×Xb is canonicallydefined, because it may be modified by adding cycles which are restrictions toX of cycles in CH>0(B)⊗ CH(X) ⊂ CH(B ×X).)

We then get the following :

Lemma 3.3.14. Assume that for a general point b ∈ B, there is a codimensionc closed algebraic subset Yb ⊂ Xb such that Lb = Hn−r

B (Xb,Q)van vanishes onXb \ Yb. If furthermore Conjecture 1.2.10 holds, there exist a closed algebraicsubset Y ⊂ X of codimension c, and a codimension n− r algebraic cycle Z ′ onX ×B X with Q-coefficients, which is supported on Y ×B Y and such that forany b ∈ B,

[Z ′b] = [∆b,van] in H2n−2rB (Xb ×Xb,Q).

Proof. This is a direct application of Proposition 3.3.10, because we knowfrom Lemma 3.3.9 that under Conjecture 1.2.10, our assumption implies thatthere exists for a very general point b ∈ B an algebraic cycle Z ′b ⊂ Yb × Yb suchthat [Z ′b] = [∆b,van] in H2n−2r(Xb ×Xb,Q).

We have next the following :

Lemma 3.3.15. With notation as in 3.3.13, let α ∈ H2n−2rB (X ×B X ,Q) be

a cohomology class whose restriction to the fibers Xb × Xb is 0. Then we canwrite

α = α1 + α2

where α1 is the restriction to X ×B X of a class β1 ∈ H2n−2rB (X × X ,Q), and

α2 is the restriction to X ×B X of a class β2 ∈ H2n−2rB (X ×X,Q).

More precisely we can take β1 ∈ ⊕i<n−rHi(X,Q)⊗L1H2n−2r−i(X ,Q), and

β2 ∈ ⊕i<n−rL1H2n−2r−i(X ,Q) ⊗ Hi(X,Q), where L stands for the Leray fil-

tration on H∗(X ,Q) with respect to the morphism π : X → B.

Proof. Consider the smooth proper morphism

(π, π) : X ×B X → B.

The relative Kunneth decomposition gives

Rk(π, π)∗Q =⊕

i+j=k

HiQ ⊗Hj

Q,


where HiQ := Riπ∗Q. The Leray spectral sequence of (π, π), which degenerates

at E2 (cf. [28]), gives the Leray filtration L on H2n−2rB (X ×B X ,Q) with graded

piecesGrl

LH2n−2rB (X ×B X ,Q) = H l

B(B, R2n−2r−l(π, π)∗Q)

=⊕

i+j=2n−2r−l

H lB(B,Hi

Q ⊗HjQ).

Our assumption on α exactly says that it vanishes in the first quotient

H0(B, R2n−2r(π, π)∗Q)

for the Leray filtration, that is equivalently, α ∈ L1H2n−2rB (X ×B X ,Q). Con-

sider now the other graded pieces

H lB(B,Hi

Q ⊗HjQ), l > 0, i + j = 2n− 2r − l.

Since l > 0, and i+ j = 2n−2r− l, we have either i < n−r or j < n−r. Let usconsider the case where i < n−r: then the Lefschetz hyperplane section theoremtells that the sheaf Hi

Q is the constant sheaf with stalk HiB(X,Q). Thus we find

that H lB(B,Hi

Q⊗HjQ) = Hi

B(X,Q)⊗H lB(B, Hj

Q), which is a Leray graded pieceof Hi

B(X,Q) ⊗ H l+jB (X ). Analyzing similarly the case where j < n − r, we

conclude that the natural map⊕

i<n−r

HiB(X,Q)⊗ L1H2n−2r−i

B (X ,Q)⊕⊕

j<n−r

L1H2n−2r−jB (X ,Q)⊗Hj

B(X,Q)

→ L1H2n−2rB (X ×B X ,Q)

is surjective. This proves the existence of the classes β1, β2.

In the case where X has trivial Chow groups, we get an extra information:

Lemma 3.3.16. With the same notations as above, assume that X has trivialChow groups and that α is the class of an algebraic cycle on X ×B X . Thenwe can choose the βi’s to be the restriction of classes of algebraic cycles onB ×X ×X.

Proof. It suffices to show that we can choose the βi’s to be the classes ofalgebraic cycles on X×X . Indeed, these classes will lift to classes on B×X×Xfor the following reason: X is a Zariski open set in the natural fibration

f : P→ X, P ⊂∏

i

Pi ×X,

P := (σ1, . . . , σr, x), σi(x) = 0, ∀1 ≤ i ≤ r.This is a fibration into products of projective spaces, because we assumed theLi’s are globally generated. It follows that X×X is as well a Zariski open set in


the corresponding fibration X ×P→ X ×X into products of projective spaces.The restriction map

CH(X ×X ×∏

i

Pi) → CH(X × P)

is then surjective, by the computation of the Chow groups of a projective bundlefibration ([86, II,9.3.2]) and the restriction map CH(X × P) → CH(X ×X ) tothe Zariski open set X ×X is also surjective. Hence the composition CH(X ×X×∏

i Pi) → CH(X×X ), and a fortiori the restriction map CH(X×X×B) →CH(X ×X ), are surjective.

It remains to show that if α is algebraic, we can choose the βi’s to be therestrictions of classes of algebraic cycles on X ×X .

We have by the proof of Lemma 3.3.15

α = β1|X×BX + β2|X×BX , (3.3.20)

where β1 ∈ H∗<n−rB (X,Q)⊗L1H∗

B(X ,Q) and β2 ∈ L1H∗B(X ,Q)⊗H∗<n−r

B (X,Q).We know that the cohomology of X is generated by classes of algebraic cycles[zi,j ] ∈ H2i

B (X,Q). Let us choose a basis [zi,j ], 2i < n − r of H∗<n−rB (X,Q).

Then we can choose cycle classes [zi,j ]∗ ∈ H2n−2r−2iB (X,Q) in such a way that

the restricted classes [zi,j ]∗|Xbform the dual basis of H∗>n−r

B (Xb,Q). We canwrite by Kunneth’s decomposition

β1 =∑

i,j

[zi,j ] ∪ γi,j , β2 =∑

i,j

γ′i,j ∪ [zi,j ], (3.3.21)

where γi,j , γ′i,j ∈ L1H∗B(X ,Q). Recalling that X ×BX ⊂ B×X×X, we denote

by p1,X the projection X ×B X → X to the first X summand and p2,X theprojection X ×B X → X to the second X summand. Let π′2 : X ×B X → Xbe the second projection. Hence π′2 is a smooth projective morphism with fiberXb over any point of Xb ⊂ X . The class α being algebraic, so is the classπ′2∗(p

∗1,X [zi,j ]∗ ∪ α). However, using (3.3.20) and (3.3.21), we have the equality

γi,j = π′2∗(p∗1,X [zi,j ]∗ ∪ α).

Indeed, we have

π′2∗(p∗1,X [zi,j ]∗ ∪ β2) = π′2∗(p

∗1,X [zi,j ]∗ ∪ (

∑

i,j

γ′i,j ∪ [zi,j ])) = 0

because γ′i,j ∈ L1H∗B(X ,Q), and

π′2∗(p∗1,X [zi,j ]∗ ∪ β1) = π′2∗(p

∗1,X [zi,j ]∗ ∪ (

∑

i,j

[zi,j ] ∪ γi,j)) = γi,j .

Similarly, π′1∗(p∗2,X [zi,j ]∗ ∪ α) = γ′i,j is algebraic on X . Thus all the γi,j ’s and

γ′i,j ’s are algebraic and so are β1, β2.


Proof of Theorem 3.3.2. We keep notation (3.3.13) and assume nowthat the vanishing cohomology Hn−r

B (Xb,Q)van is supported on a codimensionc closed algebraic subset Yb ⊂ Xb for any b ∈ B. Consider the corrected (orvanishing) diagonal Dvan, which is a codimension n − r cycle of X ×B X withQ-coefficients.

By Lemma 3.3.14, it follows that there exist a codimension c closed algebraicsubset Y ⊂ X and a codimension n− r cycle Z on X ×B X with Q-coefficients,which is supported on Y ×B Y and such that

[Zb] = [Dvan,b] = [∆b,van], ∀b ∈ B.

Thus the class [Z] − [∆b,van] ∈ H2n−2rB (X ×B X ,Q) vanishes on the fibers

Xb ×Xb.Using Lemmas 3.3.15 and 3.3.16, we conclude that there is a cycle Γ ∈

CHn−r(B ×X ×X)Q such that

[Z] = [Dvan] + [Γ|X×BX ] in H2n−2rB (X ×B X ,Q). (3.3.22)

Lemma 3.3.17. Assume Conjecture 1.2.10. If X has trivial Chow groups, thecycle class map

CH∗(X ×B X )Q → H2∗B (X ×B X ,Q)

is injective (in other words, X ×B X has trivial Chow groups).

Proof. Consider the blow-up X ×X of X×X along the diagonal. ApplyingProposition 3.3.7 and Lemma 3.3.4, X ×X has trivial Chow groups. A point ofX ×X parameterizes a couple (x, y) of points of X, together with a subschemez of length 2 of X, with associated cycle x + y. We thus have the followingnatural variety of

∏i Pi × X ×X:

Q = (σ1, . . . , σr, x, y, z), σi ∈ Pi, σi|z = 0, ∀i = 1, . . . , r.

As the Li’s are assumed to be very ample, the map Q → X ×X is a fibrationwith fiber over (x, y, z) ∈ X ×X a product of projective spaces Pi,z of codi-mension 2 in Pi. By lemma 3.3.3, Q also has trivial Chow groups. Let Q0 ⊂ Qbe the inverse image of B under the projection Q → ∏

i Pi. Then Q0 is Zariskiopen in Q, so by Lemma 3.3.5 (for which we need Conjecture 1.2.10), the cyclemap is also injective on cycles of Q0. Finally, Q0 maps naturally onto X ×B Xvia the map ∏

i

Pi × X ×X →∏

i

Pi ×X ×X.

The morphism Q0 → X ×B X being projective and dominant, we conclude byLemma 3.3.6 that the cycle map is injective on cycles of X ×B X .


The proof is then finished as follows. From the equality (3.3.22) of cohomol-ogy classes, we deduce by Lemma 3.3.17 the following equality of cycles:

Z = Dvan + Γ|X×BX in CHn−r(X ×B X )Q. (3.3.23)

We now fix b and restrict this equality to Xb ×Xb. Then we find

Zb = ∆b,van + Γ′|Xb×Xbin CHn−r(Xb ×Xb)Q,

where Γ′ ∈ CH(X ×X)Q is the restriction of Γ to b×X ×X.Recalling that ∆b,van = ∆Xb

+ Γ′′|Xb×Xbfor some codimension n − r-cycle

with Q-coefficients Γ′′ on X ×X, we conclude that

∆Xb= Zb + Γ1|Xb×Xb

, (3.3.24)

where Γ1 ∈ CHn−r(X × X,Q) and the cycle Zb is by construction supportedon Yb × Yb, with Yb ⊂ Xb of codimension ≥ c for general b.

Let z ∈ CHi(Xb)Q, with i < c. Then (Zb)∗z = 0 since we may find a cyclerationally equivalent to z in Xb and disjoint from Yb. Applying both sides of(3.3.24) to z thus gives :

z = (Γ1|Xb×Xb)∗z in CHi(Xb)Q.

But it is obvious that

(Γ1|Xb×Xb)∗ : CH(Xb)Q → CH(Xb)Q

factors through jb∗ : CH(Xb)Q → CH(X)Q. Now, if z is homologous to 0 onXb, jb∗(z) is homologous to 0 on X, and thus it is rationally equivalent to 0 onX because X has trivial Chow groups. Hence we proved that the cycle mapwith Q-coefficients is injective on CHi(Xb)Q for i < c, which concludes the proofof the theorem.

3.3.5 Further applications

Complete intersections with group action

Theorem 3.3.2 applies to general complete intersections in projective space.The relation (3.1.1) gives the Hodge coniveau for them, hence conjecturally thegeometric coniveau c, according to the generalized Hodge conjecture 1.2.21. Theconclusion is that solving the generalized Bloch conjecture for them is equivalentto solving a strong form of the generalized Hodge conjecture, implied by thegeneralized Hodge conjecture together with the standard Lefschetz conjecture.

There are interesting variants coming from the study of complete intersec-tions Xb of r hypersurfaces in projective space Pn, or in a product of projec-tive spaces, invariant under a finite group action. Let G acts on Xb, and letχ : G → Z/2Z = 1,−1 be a character of G. Then consider the sub-Hodgestructure

Lχ = α ∈ Hn−r(Xb,Q)prim, g∗α = χ(g)α, ∀g ∈ G ⊂ Hn−r(Xb,Q)prim.


In general, it has a larger coniveau than Xb. For example if Xb is a quinticsurface in P3, defined by an invariant polynomial under the linearized groupaction of G ∼= Z/5Z with generator g on P3 given by

g∗Xi = ζiXi, i = 0, . . . , 3,

where ζ is a nontrivial 5-th root of unity, then H2(S,Q)inv has no (2, 0)-part,hence is of coniveau 1, while H2,0(S) 6= 0 so the coniveau of H2(S,Q)prim is 0.The quotient surfaces S/G is a quintic Godeaux surface (cf. [83]).

Note that the Hodge structure Lχ corresponds to the projector 1|G|

∑g∈G χ(g)g∗

acting on L, and it is given by the action of the n− r-cycle

Γχ :=1|G|

∑

g∈G

χ(g)∆b,van,g,

where ∆b,van,g = (Id, g)∗(∆b,van) ∈ CHn−r(Xb ×Xb)Q. The generalized Blochconjecture 2.2.4 (extended to motives) predicts the following :

Conjecture 3.3.18. Assume Lχ has coniveau ≥ c. Then the cycle map isinjective on CH(Xb)

χQ for i < c.

If χ is the trivial character, this conjecture is essentially equivalent to theprevious one by considering X/G or a desingularization of it. Even in this case,one needs to make assumptions on the linearized group action in order to applythe same strategy as in the proof of Theorem 3.3.2. The case of non trivialcharacter cannot be reduced to the previous case.

In order to apply a strategy similar to the one applied for the proof ofTheorem 3.3.2, we need some assumptions. Indeed, if the group G is too big,like the automorphisms group of the Fermat hypersurface, there are to fewinvariant complete intersections to play on the geometry of the universal familyX → B of G-invariant complete intersections.

In any case, what we get mimicking the proof of Theorem 3.3.2 is the follow-ing: X is as before a smooth projective variety of dimension n with trivial Chowgroups and G is a finite group acting on X. We study complete intersectionsXb ⊂ X of r G-invariant ample hypersurfaces Xi ∈ |Li|G : Let G acts via thecharacter χi on the considered component of |Li|G. The basis B parameterizingsuch complete intersections is thus a Zariski open set in

∏i P(H0(X,Li)χi). As

before we denote by X → B the universal complete intersection.

Theorem 3.3.19. Assume(i) The variety X ×B X has trivial Chow groups.(ii) The Hodge structure on Hn−r

B (Xb,Q)χvan is supported on a closed alge-

braic subset Yb ⊂ Xb of codimension c. (According to Conjecture 1.2.21, this issatisfied if the Hodge coniveau of Hn−r

B (Xb,Q)χvan is ≥ c.)

(iii) Conjecture 1.2.10 holds for codimension n− r cycles.Then the cycle map CHi(Xb)

χQ → H2n−2r−2i

B (Xb,Q)χ is injective for i < cand for any b ∈ B.


Remark 3.3.20. In the case where Xb are surfaces with h2,0(Xb)χ = 0, theassumption (ii) and (iii) are automatically satisfied, by Lefschetz theorem on1, 1)-classes. We thus get an alternative proof of the main theorem of [83], wherethe Bloch conjecture is proved for the general Godeaux surfaces (quotients ofquintic surfaces by a free action of Z/5Z, or quotients of complete intersectionsof four quadrics in P6 by a free action of Z/8Z).

In the case of threefolds Xb of Hodge coniveau 1, we can also conclude thatCH0(Xb)

χ0 = 0 if (i) is satisfied and the generalized Hodge conjecture is satisfied

by the coniveau 1 Hodge structure on H3(Xb,Q)χ. Indeed, we used conjecture1.2.10 in the proof in two places: The first place is in the proof of Lemma 3.3.9,which says that if a certain Hodge structure L ⊂ H∗

B(Xb,Q) is supported ona codimension c closed algebraic subset Yb, the corresponding projector has aclass which comes from the class of a cycle supported in Yb × Yb. This willbe satisfied if dim Xb = 3, L ⊂ H3

B(X,Q)χ supported on Yb × Yb because weknow that the degree 6 Hodge class of the projector πL is supported on thecodimension 2 closed algebraic subset Yb × Yb (or rather a desingularization ofit), so that we can apply Lemma 1.2.12. The second place is in the proof ofLemma 3.3.17. However, in the threefold case, it is possible to prove it directlywithout using Conjecture 1.2.10 (cf. [94]).

This way the second result of [83] (quintic hypersurfaces with involutions)and the main application of [69] (3-dimensional complete intersection in weightedprojective space) are reproved : in both cases we are reduced to prove the gener-alized Hodge conjecture for the coniveau 1 Hodge structure on their cohomologyof degree 3.

Application to self-products

Let Y be a smooth projective variety. There is a natural surjective map

CH0(Y )⊗ CH0(Y ) → CH0(Y × Y ),

sending (z, z′) to pr∗1z · pr∗2z′. More generally one can study the product map:

CH0(Y )⊗m → CH0(Y m)

which is defined similarly, and is compatible with the action of the symmetricgroup Sm on both sides.

This study has been undertaken in [49], [84] and a variant of it (where theemphasis is on cycles of given codimension, instead of cycles of given dimension)is studied in [50].

Notice that if we fix a point o ∈ Y , we always have an injection

CH0(Y ) → CH0(Y × Y ), z 7→ z × o = pr∗1z · pr∗2o,

because the composition of this map with pr1∗ : CH0(Y × Y ) → CH0(Y ) isthe identity. We can argue similarly after exchanging factors. The conclusionof this is that the interesting map is the following one:

CH0(Y )hom ⊗ CH0(Y )hom → CH0(Y × Y )hom (3.3.25)


This map is rather mysterious. It is proved in [84] that if Y is a surface andthis map, or only the symmetric part of it, is trivial, then the surface Y (whichnecessarily has h2,0(Y ) = 0) satisfies the Bloch conjecture.

Let us spell-out what predicts the generalized Bloch conjecture (adapted tomotives) for the map (3.3.25), or rather its antisymmetric or symmetric versions.

We start with the following Lemma.

Lemma 3.3.21. Let H be a Hodge structure of weight m. Then for k > hm,0 :=dim Hm,0, the Hodge structure of weight km on

∧kL has coniveau ≥ 1. In

particular, if hm,0 = 1, the Hodge structure of weight 2m on∧2

H has coniveau≥ 1.

Proof. Indeed, the (km, 0)-piece of the Hodge structure on∧k

H is equalto

∧kHm,0, hence it is 0 for k > hm,0.

Let Y be smooth projective variety. Assume that Hi,0(Y ) = 0 for i 6= 0, m(this will be the case if Y is a m-dimensional complete intersection of amplehypersurfaces in a projective variety with trivial Chow groups). Conjecture 2.2.4(or rather its generalization to motives) together with Lemma 3.3.21 predictsthe following (see below for more detail):

Conjecture 3.3.22. Assume Y satisfies the above assumption and has hm,0(Y ) =1. Then, for any z, z′ ∈ CH0(Y ) with deg z = deg z′ = 0, one has z×z′−z′×z =0 in CH0(Y ×Y ) for m even and z× z′+ z′× z = 0 in CH0(Y ×Y ) for m odd.

The case m = 2 is particularly interesting, as noticed in [84]. In this case,we indeed have :

Lemma 3.3.23. Let (H, Hp,q) be a weight 2 Hodge structure of K3 type, namelyh2,0 = 1. Then the Hodge structures on

∧2kH all have niveau ≤ 2 (that is

coniveau ≥ k − 1).

Proof. Write H = H2,0⊕

H1,1 ⊕H0,2. Then

k∧H = H2,0 ⊗

k−1∧H1,1 ⊕ (

k∧H1,1 ⊕H2,0 ⊗H0,2 ⊗

k−2∧H1,1)⊕

k−1∧H1,1 ⊗H0,2

is the Hodge decomposition of∧k

H, whose first nonzero term is of type (k +1, k − 1).

When k > dim H, we of course have that the Hodge structure on∧k

H istrivial. Applying these observations to the case where H = H2

B(S,Q) where S isan algebraic K3 surface, we find that Conjecture 2.2.4 (or rather, its extensionto motives) predicts the following (cf. [84]):

Conjecture 3.3.24. (i) Let S be an algebraic K3 surface. Then for any k ≥ 2,and i ≤ k−2, the projector πalt =

∑σ∈Sk

(−1)ε(σ)σ ∈ CH2k(Sk×Sk) composedwith the Chow-Kunneth projector π⊗k

2 (cf. [63]) acts as 0 on CHi(Sk)Q fori ≤ k − 2.

(ii) For k > b2(S), this projector is identically 0.


Note that (ii) above is essentially Kimura’s finite dimensionality conjecture[49] and applies to any regular surface. One may wonder whether it could beattacked by the methods used for the proof of Theorem 3.3.2 for the case ofquartic K3 surfaces. The question would be essentially to study whether thefibered product X 2k/B of the universal such K3 surface has trivial Chow groups.For small k this is easy, but we would need to know this in the range k ≥ 44 inorder to prove the Kimura conjecture. This seems to be very hard.

The fact that this is true for small k (see below) shows that Conjecture 3.3.24is implied by the generalized Hodge conjecture for the self-products Sk and theconiveau k − 1 Hodge structures

∧kH2

B(S,Q) ⊂ H2kB (Sk,Q).

Example 3.3.25. (Abelian varieties) Conjecture 3.3.22 works as well with va-rieties replaced by motives. Consider for example an abelian variety A of di-mension g. The Kunneth conjecture 1.2.8 holds for them, and a stronger versionof it, namely there is a decomposition of the diagonal of A as a sum of projec-tors πi in CH(A×A)Q, where the class of pi is the Kunneth component δi (cf.[63]). Consider the motive Mg(A) := (A, πg). It has hi,0 = 0 for i 6= g andhg,0 = hg,0(A) = 1.

What is thus predicted in this case is the fact that the morphism

CH0(Mg(A))⊗ CH0(Mg(A)) → CH0(Mg(A)⊗Mg(A))

is (−1)g-symmetric. This is indeed true using the Beauville decomposition [7]which is also used in [30] to construct the projectors πi. This decomposition isa splitting of the Chow groups of any abelian variety (in our case A and A×A)into a sum of eigenspaces under the action of homotheties

µi : A → A, a 7→ ia.

The Chow group is identified (by the construction of πg) to the piece of CH0(A)Qwhere µi acts by multiplication by iq. It is also generated by products D1·. . .·Dg,where the Di’s are divisors homologous to 0 on A (cf. [11], [7]). We now follow[84]. It follows from the above that the image of the morphism

CH0(Mg(A))⊗ CH0(Mg(A)) → CH0(Mg(A)⊗Mg(A))

consists of 0-cycles generated by products of divisors

pr∗1D1 · . . . · pr∗1Dg · pr∗2D′1 · . . . · pr∗2D′

g (3.3.26)

where the Di’s and D′i’s are divisors homologous to 0 on A. We want to show

that the involution τ : A×A → A×A, (a, b) 7→ (b, a) acts by (−1)g on products(3.3.26). Consider the map

σ : A×A → A×A, (a, b) 7→ (a + b, a− b).

This is an isogeny of A × A, since σ σ = 2IdA×A. It follows that it inducesan isomorphism (in fact an homothety) at the level of Chow groups with Q-coefficients. On the other hand, σ τ σ = (2IdA,−2IdA). It thus suffices toshow that (2IdA,−2IdA) acts by (−1)g on products (3.3.26), which is obviousbecause −IdA acts by −Id on Pic0(A).


Let us state explicitely what give the arguments of the proof of Theorem3.3.2 in the case of general Calabi-Yau complete intersections and for k = 2. LetXb be a smooth Calabi-Yau complete intersection of dimension m in projectivespace Pn. Let ∆b,van ∈ CHm(Xb×Xb)Q be the corrected diagonal, whose actionon H∗

B(Xb,Q) is the projection on HmB (Xb,Q)van. On Xb×Xb×Xb×Xb, there

is the induced 2m-cycle

∆b,van,2 := p∗13∆b,van · p∗24∆b,van,

where pij is the projection from X4b to the product X2

b of its i-th and j-th factor.The action on ∆b,van,2 seen as a self-correspondence of X2

b on H∗(X2b ,Q) is the

orthogonal projector on

p∗1HmB (Xb,Q)van ⊗ p∗2H

mB (Xb,Q)van ⊂ H2m

B (Xb ×Xb,Q).

If instead of ∆b,van,2, we consider

∆τb,van,2 := p∗14∆b,van · p∗23∆b,van,

then the action on ∆b,van,2 seen as a self-correspondence of X2b on H∗

B(X2b ,Q)

is the composition of the previous projector with the permutation

τ∗ : HmB (Xb,Q)van ⊗Hm

B (Xb,Q)van → HmB (Xb,Q)van ⊗Hm

B (Xb,Q)van

exchanging summands. Note that the inclusion

HmB (Xb,Q)van ⊗Hm

B (Xb,Q)van ⊂ H2mB (Xb ×Xb,Q)

sends the antiinvariant part on the left to the antiinvariant part under τ onthe right if m is even, and to the invariant part under τ on the right if m iseven. This is due to the fact that the cup-product on cohomology is gradedcommutative.

Hence we conclude that

∆]b,van,2 := ∆b,van,2 −∆τ

b,van,2

acts on H∗B(X2

b ,Q) as twice the projector onto∧2

HmB (Xb ×Xb,Q)van if m is

even, and that∆inv

b,van,2 := ∆b,van,2 + ∆τb,van,2

acts on H∗B(X2

b ,Q) as twice the projector onto∧2

HmB (Xb ×Xb,Q)van if m is

odd.In both cases, using Lemma 3.3.21, we get that this is twice the orthogonal

projector associated to a sub-Hodge structure of coniveau ≥ 1.Restricting to the case of Calabi-Yau hypersurfaces in Pn, (so m = n − 1),

an easy adaptation of the proof of Theorem 3.3.2 gives now:


Theorem 3.3.26. Assume Conjecture 1.2.10 is true and the generalized Hodgeconjecture holds for the coniveau 1 Hodge structure on

∧2Hn−1

B (Xb×Xb,Q)van ⊂H2n−2

B (Xb×Xb,Q), where Xb is a very general Calabi-Yau hypersurface in pro-jective space. Then the general such Xb has the following property:

(i) If n − 1 is even, for any two 0-cycle z, z′ of degree 0 on Xb, we havez × z′ − z′ × z = 0 in CH0(X ×X).

(ii) If n − 1 is odd, for any two 0-cycle z, z′ of degree 0 on Xb, we havez × z′ + z′ × z = 0 in CH0(X ×X).

Note that the proof has to be adapted, because we are interested into theself-correspondence ∆]

b,van,2 or ∆invb,van,2 of Xb×Xb, which is a cycle in X4

b . Thismeans that we have to work with cycles on the the fourth fibered self-productof the universal family X → B.

Chapter 4

On the Chow ring of K3surfaces and hyper-Kahlermanifolds

This chapter is devoted to a completely different application of Theorem 2.1.1.We will consider varieties whose Chow ring has rather special properties. Thisincludes abelian varieties, K3 surfaces, and Calabi-Yau hypersurfaces in pro-jective space. For K3 surfaces S, it was discovered in [10] that they have acanonical 0-cycle o of degree 1 with the property that the product of two divisorsof S is a multiple of o in CH0(S). We extend in [95] this result to Calabi-Yau hypersurfaces in projective space. Another feature is a decomposition inCH(X ×X ×X)Q of the small diagonal ∆ ⊂ X ×X ×X that was establishedfor K3 surfaces in [10] for K3 surfaces, and is partially extended in [95] toCalabi-Yau hypersurfaces.

Finally, we use this decomposition and the spreading principle (Theorem2.1.1) to show that for families of smooth projective K3 surfaces π : X → B,there is a decomposition isomorphism

Rπ∗Q ∼= ⊕Riπ∗Q[−i]

which is multiplicative (that is compatible with cup-product on both sides) overa nonempty Zariski dense open set of B.

Numerous examples show that this statement, which is also true for familiesof abelian varieties, is rarely satisfied.

4.1 Canonical ring of a K3 surface

The following Theorem is proved in [10]:

85

86 CHAPTER 4. CHOW RING OF K3 SURFACES

Theorem 4.1.1. (Beauville-Voisin 2004) Let S be a K3 surface, Di ∈ CH1(S)be divisors on S and nij be integers. Then if the 0-cycle

∑i,j nijDiDj ∈ CH0(S)

is cohomologous to 0 on S, it is equal to 0 in CH0(S).

Proof. It suffices to prove that there is a zero cycle o of degree 1, with theproperty that for any two line bundles L, L′ on S,

L · L′ = deg (L · L′) o in CH0(S). (4.1.1)

The cycle o is defined to be the class of any point of S contained in a(singular) rational curve C ⊂ S. We claim that this cycle does not depend onthe choice of rational curve. This follows from the fact that there are rationalcurves C0 in any ample linear system on S. If C, C ′ are two rational curves onS, any point in the intersection of C and C0 is rationally equivalent to any pointsupported on C or on C0, and also to any point supported on C ′ if C ′ ∩ C0 isnonempty. As C0 is ample, the intersections C0 ∩C, C0 ∩C ′ are nonempty, andthis concludes the proof of the claim.

The proof of (4.1.1) is obtained by reducing to the case where L and L′ areample line bundles. Then there are rational curves C, C ′ in | L | and | L′ |. Theintersection L · L′ = C · C ′ is then by definition proportional to o.

The cycle o has the following quite remarkable property (which makes itintrinsically defined):

Proposition 4.1.2. (Beauville-Voisin [10]) We have the following equality:

c2(TS) = 24 o in CH0(S).

Proposition 4.1.2 will be a consequence of the following result, which willbe proved in section 4.2. Let ∆ = (x, x, x), x ∈ S be the small diagonal inS × S × S.

Theorem 4.1.3. (Beauville-Voisin [10]) We have the following equality inCH4(S × S × S)Q

∆ = ∆12 × o3 + (perm.)− (o1 × o2 × S + (perm.)) (4.1.2)

Here ∆ ⊂ S × S × S is the small diagonal (x, x, x), x ∈ S. The classo ∈ CH0(S) is the class of any point as above and the oi’s are its pull-backin CH2(S × S × S)Q via the various projections. The cycle ∆12 × o3 is thenthe class of the algebraic subset (x, x, o), x ∈ S of S × S × S. The terms“+(perm.)” mean that we sum over the permutations of 1, 2, 3. As our cycles∆12×o3 and o1×o2×S are invariant under the transposition exchanging 1 and2, there are only three terms of each sort in these sums.

Proof of Proposition 4.1.2. Let us restrict equality (4.1.2) to

(jS , Id)(S × S) ⊂ S × S × S,

4.1. CANONICAL RING OF A K3 SURFACE 87

where jS : S → S × S is the inclusion of the diagonal. We get the followingequality in CH4(S × S)Q:

(jS , Id)∗∆ = (∆12 × o3)|(jS ,Id)(S×S) + (∆13 × o2)|(jS ,Id)(S×S) (4.1.3)+(o1 ×∆23)|(jS ,Id)(S×S) − (o1 × o2 × S)|(jS ,Id)(S×S)

−(o1 × S × o3)|(jS ,Id)(S×S) − (S × o2 × o3)|(jS ,Id)(S×S).

The left hand side is clearly equal to the self-intersection ∆2S ∈ CH4(S × S)Q,

that is to jS∗c2(TS). The right hand side is equal to

c2 × o + (o, o) + (o, o)− (o, o)− (o, o) = c2 × o.

Hence we obtain the equality:

jS∗c2(TS) = c2 × o in CH0(S × S)Q.

Applying the second projection to this equality, we conclude that

c2(TS) = deg c2(TS) o = 24 o in CH0(S)Q,

and the equality is in fact true as well in CH0(S) since CH0(S) as no torsionby Roitman’s theorem [73].

Remark 4.1.4. Theorem 4.1.3 also implies Theorem 4.1.1, because we can seethe small diagonal ∆ as a correspondence between S × S and S. Then we have

∆∗(∑

i,j

nijpr∗1Di · pr∗2Dj) =∑

i,j

nijDi ·Dj in CH0(S)Q

and the left hand side is computed using formula (4.1.2). The conclusion is thatthe cycle

∑i,j nijDiDj is a multiple of o in CH0(S)Q and thus it is trivial if it

is homologous to 0. However, one should be warned that Theorem 4.1.1 is infact used in the proof of Theorem 4.1.3 (see below) so that it cannot be seen asa consequence of it.

We conclude with another remarkable property of the cycle o.

Lemma 4.1.5. Let jS : S → S × S be the diagonal inclusion. Then for anyline bundle L ∈ PicS, we have

jS∗L = L× o + o× L in CH1(S × S).

Proof. Both sides are Z-linear in L. We use the fact that Pic S is generatedby OS(C), where C is a (singular) rational curve in S. For the normalizationC ∼= P1 of C, we have

∆C = C × o1 + o1 × C in CH1(C × C) (4.1.4)

for any point o1 of C. Let n : C → S be the natural map. By definition of o, wehave n∗o1 = o in CH0(S), and applying (n, n)∗ to (4.1.4) gives thus the desiredresult.


4.1.1 Other hyper-Kahler manifolds

Recall that (irreducible) hyper-Kahler manifolds are simply connected projectivecomplex varieties with H2,0(X) = Cη, where η is an everywhere nondegenerateholomorphic 2-form on X. One famous series of examples is constructed in [6]:one considers the Hilbert scheme S[n] of length n subschemes of an algebraic K3surface S. Such hyper-Kahler varieties admit projective deformations which arenot obtained by the same construction, but they are not well understood exceptin a few cases, namely the three different families of hyper-Kahler fourfoldsconstructed in the papers [9], [24], [46]. The varieties constructed by Beauvilleand Donagi are obtained as Fano varieties of lines of smooth cubic fourfolds inP5.

In [8], Beauville conjectured that a result similar to Theorem 4.1.1 holds foralgebraic hyper-Kahler varieties:

Conjecture 4.1.6. (Beauville 2007) Let Y be an algebraic hyper-Kahler vari-ety. Then any polynomial cohomological relation

P ([c1(Li)]) = 0 in H∗(Y,Q), Li ∈ PicY

already holds at the level of Chow groups :

P (c1(Li)) = 0 in CH(Y )Q.

He proved in [8] this conjecture in the case of the second and third punctualHilbert scheme of an algebraic K3 surface.

In the paper [91], we stated the following more general conjecture concern-ing the Chow ring of an (irreducible algebraic) hyper-Kahler variety. Namely, asynthesis of Theorem 4.1.1 and Proposition 4.1.2 is the statement that any poly-nomial relation between the cohomology classes [c2(TS)], [c1(Li)] in H∗(S,Q),already holds between the cycles c2(TS), c1(Li) in CH(S).

Conjecture 4.1.7. (Voisin 2008) Let Y be an algebraic hyper-Kahler variety.Then any polynomial cohomological relation

P ([c1(Lj)], [ci(TY ))]) = 0 in H2k(Y,Q), Lj ∈ Pic Y

already holds at the level of Chow groups :

P (c1(Lj), ci(TY )) = 0 in CHk(Y )Q.

The following results are proved in [91]:

Theorem 4.1.8. (1) Conjecture 4.1.7 holds for Y = S[n], where S[n] is theHilbert scheme of length n subschemes of an algebraic K3 surface S, in therange n ≤ 2b2(S)tr + 4.

(2) Conjecture 4.1.7 is true for any k when Y is the Fano variety of lines ofa cubic fourfold.


In item (1) above, the number b2(S)tr is equal to b2(S)− ρ(S) = 22− ρ(S),where ρ(S) is the Picard number of S. More importantly, the method of prooffor (1) shows Theorem 4.1.10 below.

We also consider the following conjecture, involving only the usual productsSm of S.

Conjecture 4.1.9. (Voisin 2008 [91]) Let S be an algebraic K3 surface. Forany integer m, let P ∈ CH(Sm)Q be a polynomial expression in

pr∗i c1(Ls), Ls ∈ Pic S, pr∗j o, pr∗kl∆S ,

where the pri, resp. prkl are the projections from Sm to S, resp. S × S.Then if [P ] = 0 in H∗(Sm,Q), we have P = 0 in CH(Sm)Q.

Note that by Theorems 4.1.1 and 4.1.3, we may assume in the above Con-jecture 4.1.9 that the polynomial P involves only monomials

Π(i,j,k,l)∈1,...,m4pr∗i c1(Ls) · pr∗j o · pr∗kl∆S

with four different indices i, j, k, l.

Theorem 4.1.10. Conjecture 4.1.7 is implied by Conjecture 4.1.9.

Let us give an idea of the proof of these theorems. It mainly uses theinductive method of [31] and the result of [23] computing (additively) the Chowgroups of CH(S[m])Q in terms of Chow groups of strata of S(m) (see below).

This inductive method necessitates proving a more general statement (The-orem 4.1.11):

There are two natural vector bundles on S[n], namely O[n] on the one hand,which is defined as R0p∗OΣn , where

Σn ⊂ S[n] × S, p = pr1 : Σn→S[n]

is the incidence subscheme, and the tangent bundle Tn of S[n] on the other hand.It is not clear that the Chern classes of O[n] can be expressed as polynomials inc1(O[n]) and the Chern classes of Tn.

Theorem 4.1.11. Let n ≤ 2b2(S)tr + 4, and let P ∈ CH(S[n])Q be any poly-nomial expression in the variables

c1(L), L ∈ PicS ⊂ PicS[n], ci(O[n]), cj(Tn) ∈ CH(S[n])Q.

Then if P is cohomologous to 0, we have P = 0 in CH(S[n])Q.

This implies Theorem 4.1.8 for the n-th Hilbert scheme of K3 surface S withn ≤ 2b2(S)tr + 4, because we have c1(O[n]) = −δ, where 2δ ≡ E is the class ofthe exceptional divisor of the resolution S[n] → S(n), and it is well-known thatPicS[n] is generated by Pic S and δ.

The proof of this theorem uses the following proposition:


Proposition 4.1.12. Let P ∈ CH(Sm)Q be a polynomial expression in thevariables

pr∗i (124

c2(T )) = pr∗i o, pr∗j c1(Ls), Ls ∈ PicS, pr∗kl∆S , k 6= l,

where ∆S ⊂ S×S is the diagonal. Assume that one of the following assumptionsis satisfied:

1. m ≤ 2b2(S)tr + 1.

2. P is invariant under the action of the symmetric group Sm−2 acting onthe m− 2 first indices.

Then if P is cohomologous to 0, it is equal to 0 in CH(Sm)Q.

Using the results of [10] described in the previous sections (Theorem 4.1.1and Proposition 4.1.2), this proposition is a consequence of the following lemma(cf. [91] for the proof):

Lemma 4.1.13. The polynomial relations [P ] = 0 in the cohomology ringH∗(Sm), satisfying one of the above assumptions on m, P , are all generated(as elements of the ring of all polynomial expressions in the variables above) bythe following polynomial relations, the list of which will be denoted by (*) :

1. [pr∗i (c1(L)) · pr∗i o] = 0, L ∈ Pic S, [pr∗i (o) · pr∗i (o)] = 0.

2. [pr∗i (c1(L)2 − [c1(L)]2o)] = 0, L ∈ PicS.

3. [pr∗ij(∆S .p∗1o − (o, o))] = 0, where p1 here is the first projection of S × Sto S, and (o, o) = p∗1o · p∗2o.

4. [pr∗ij(∆S .p∗1c1(L) − c1(L) × o − o × c1(L))] = 0, L ∈ PicS, where p1 hereis the first projection of S × S to S, and c1(L)× o = p∗1c1(L) · p∗2o.

5. [pr∗ijk(∆− p∗12∆S · p∗3o− p∗1o · p∗23∆S − p∗13∆S · p∗2o + p∗12(o, o) + p∗23(o, o) +p∗13(o, o))] = 0.

6. [pr∗ij∆S ]2 = 24pr∗ij(o, o) = 24pr∗i o · pr∗j o.

In 5, ∆ is as before the small diagonal of S3 and the pi, pij are the variousprojections from S3 to S, S × S respectively. Note that ∆ can be expressed asp∗12∆S · p∗23∆S . Furthermore we have

pr∗ij p∗1 = pr∗i , pr∗ijk p∗12 = pr∗ij , pr∗ijk p∗i = pr∗i .

Thus all the relations in (*) are actually polynomial expressions in the variables

[pr∗i o], [pr∗j c1(L)], L ∈ PicS, [pr∗kl∆S ], k 6= l.

Assuming this Lemma, we conclude that for m ≤ 2b2(S)tr+1, all polynomialrelations [P ] = 0 in the variables pr∗i o, pr∗j c1(L), L ∈ PicS, pr∗kl∆S , k 6= l which


hold in H∗(Sm,Q) also hold in CH(Sm)Q, because we know by Theorem 4.1.1,Proposition 4.1.2 and Theorem 4.1.3 that the cohomological relations listed in(*) hold in CH(Sm)Q. In fact, (apart from the relations 1 and 3 which obviouslyhold in CH(Sm)Q), these relations are pulled-back, via the maps pri, resp. prij ,resp. prijk, from relations in CH(S)Q, resp. CH(S2)Q, resp. CH(S3)Q, whichare established there.

Similarly, for any m, the same conclusion holds for polynomial relationsinvariant under Sm−2.

This concludes the proof of Proposition 4.1.12.

In order to sketch the proof of Theorem 4.1.11, let us introduce the followingnotation: let

µ = µ1, . . . , µm, m = m(µ),∑

i

| µi |= n

be a partition of 1, . . . , n. Such a partition determines a partial diagonal

Sµ∼= Sm ⊂ Sn,

defined by the conditions

x = (x1, . . . , xn) ∈ Sµ ⇔ xi = xj if i, j ∈ µl, for some l.

Consider the quotient map

qµ : Sm ∼= Sµ → S(n),

and denote by Eµ the following fibered product:

Eµ := Sµ ×S(n) S[n] ⊂ Sm × S[n].

We view Eµ as a correspondence between Sm and S[n] and we will denote asusual by E∗

µ : CH(S[n])Q → CH(Sm)Q the map

α 7→ pr1∗(pr∗2(α) · Eµ).

Let us denote by Sµ the subgroup of Sm permuting only the indices i, j forwhich the cardinalities of µi, µj are equal. The group Sµ can be seen as thequotient of the global stabilizer of Sµ in Sn by its pointwise stabilizer. In thisway the action of Sµ on Sµ

∼= Sm is induced by the action of Sn on Sn.We have the following result:

Proposition 4.1.14. Let P ∈ CH(S[n])Q be a polynomial expression in thevariables ci(O[n]), cj(Tn). Then for any µ as above, E∗

µ(P ) ∈ CH(Sm) is apolynomial expression in pr∗so, pr∗lk∆S. Furthermore, E∗

µ(P ) is invariant underthe group Sµ.

Note that the last statement is obvious, since Sµ leaves invariant the corre-spondence Eµ ⊂ Sµ × S[n].


The proof of this Proposition is rather painful and we refer to [91] for thedetail. This is here that we use the Ellingsrud-Gottsche-Lehn method.

Admitting this proposition, we give now the proofs of the theorems, following[91].

Proof of Theorem 4.1.11. From the work of De Cataldo-Migliorini [23],we know that the map

(E∗µ)µ∈Part(1,...,n) : CH(S[n])Q → ΠµCH(Sm(µ))Q

is injective. Let now P ∈ CH(S[n])Q be a polynomial expression in c1(L), L ∈PicS ⊂ Pic S[n], ci(O[n]), cj(Tn) ∈ CH(S[n])Q. Note first that for L ∈ PicS,and for each µ, the restriction of pr∗2L to Eµ ⊂ Sµ×S[n] is a pull-back pr∗1Lµ|Eµ

,where Lµ ∈ PicSµ = Pic Sm is equal to L⊗|µ1|£ . . .£L⊗|µm|. This follows fromthe fact that L is the pull-back of a line bundle on S(n). Note that Lµ is invariantunder Sµ.

Thus it follows from Proposition 4.1.14 and the projection formula that foreach partition µ, E∗

µ(P ) is a polynomial expression in pr∗i c1(L), pr∗ko, pr∗lm∆which is invariant under the group Sµ.

Now, if P is cohomologous to 0, each E∗µ(P ) is cohomologous to 0. Let us

now verify that the assumptions of Proposition 4.1.12 are satisfied. Recall thatwe assume n ≤ 2b2(S)tr +4. If m(µ) ≤ 2b2(S)tr +1, Proposition 4.1.12 applies.Otherwise, m(µ) ≥ 2b2(S)tr + 2 and, as n ≤ 2b2(S)tr + 4, it follows that thepartition µ contains at most two sets of cardinality ≥ 2. Thus the group Sµ

contains in this case a group conjugate to Sm(µ)−2. Proposition 4.1.12 thusapplies, and gives E∗

µ(P ) = 0 in CH(Sµ)Q, for all µ.It follows that P = 0 by the result of De Cataldo-Migliorini. This concludes

the proof of Theorem 4.1.11.

To conclude, let us notice that Proposition 4.1.14 and the end of the proofof Theorem 4.1.11 prove as well Theorem 4.1.10.

4.2 A decomposition of the small diagonal

We give in this section the proof of Theorem 4.1.3. We first recall the statement:

Theorem 4.2.1. (Beauville-Voisin [10]) We have the equality of cycles inCH4(S × S × S)Q :

∆ = ∆12 × o3 + (perm.)− (o1 × o2 × S + (perm.)) (4.2.5)

The proof of Theorem 4.2.1 will use the analogous result for elliptic curves:

Proposition 4.2.2. Let E be an elliptic curve, l a divisor of degree d on E,then:

(i) For any x ∈ E, we have the following equality in CH0(E × E):

d2(x, x) = d(x× l + l × x)− l × l. (4.2.6)

4.2. A DECOMPOSITION OF THE SMALL DIAGONAL 93

(ii) We have the following equality of cycles in CH2(E × E × E)Q:

d2∆ = d[∆12 × l3 + (perm.)]− (l1 × l2 × E + (perm.)) (4.2.7)

where again ∆ ⊂ E ×E ×E is the small diagonal, and l1× l2×E := pr∗1l · pr∗2let cetera.

Proof. (i) Indeed, both cycles in (4.2.6) are symmetric with respect to theinvolution ι exchanging factors of E×E. Hence (up to torsion) they come fromcycles in CH0(E×E/ι). The group CH0(E×E/ι) is representable, because thequotient E×E/ι = E(2) is a P1-bundle over E. In other words, the Chow groupCH0(E(2)) is an extension of Z by the Albanese variety of S(2)E. In order tocheck (4.2.6), (at least up to torsion), it thus suffices to verify that both sideshave the same degree and that their difference has a trivial Albanese invariant,which is elementary. This proves a priori the result only up to torsion, but asRoitman’s theorem [73] says that the Albanese map is injective on the torsionof CH0, this gives as well the complete result.

(ii) We use (i) and apply Corollary 2.1.5 to the small diagonal of E seen asa correspondence between E and E ×E. We thus deduce that there are pointspi of E and cycles Zi ∈ CH1(E ×E)Q such that the following equality holds inCH2(E × E × E)Q):

d2∆ = [d[∆12 × l3 + (perm.)]− (l1 × l2 × E + (perm.)) +∑

i

pi × Zi. (4.2.8)

Using the fact that the 1-cycle d2∆ − [d[∆12 × l3 + (perm.)] + (l1 × l2 × E +(perm.)) is invariant under the symmetric group S3, we conclude that the 1-cycle

∑i pi × Zi in CH2(E ×E ×E)Q is also invariant under the group S3. It

follows that it is cohomologous to 0 and Abel-Jacobi equivalent to 0. But theDeligne cycle class (with value in Deligne cohomology H4

D(E×E×E,Q(2))) isinjective on the invariant part CH2(E×E×E)S3

Q , because E(3) is a P2-bundleover E.

Note that we can deduce from Proposition 4.2.2,(i) the following propertysatisfied by the cycle o (which is a weak version of Theorem 4.2.1) :

Corollary 4.2.3. For any 0-cycle z ∈ CH0(S), one has

iS∗(z) = z × o + o× z − deg z (o, o) in CH0(S × S),

where iS : S → S × S is the inclusion of the diagonal of S × S.

Proof. It suffices to prove the result when z is a point of S. We know that Sis swept-out by (singular) elliptic curves, that is curves C whose normalizationE is a smooth elliptic curve. A general point x of S belongs to a smooth point ofsuch a curve C, and we will denote by x the corresponding point of E. Denotingby j : E → S the natural map, so that x = j(x), we now choose for line bundle


l on E any line bundle of the form j∗L, where L ∈ PicS has non zero degree don C. We apply (4.2.6) on E and then apply j∗. We then get

d2(x, x) = d(x× L · C + L · C × x)− (j, j)∗(j∗L× j∗L) in CH0(S × S).

We know by Theorem 4.1.1 that L · C = d o in CH0(S). It follows as well that

j∗(j∗L) = d o in CH0(S).

Hence we conclude that

d2(x, x) = d2(x× o + o× x)− d2(o, o) in CH0(S × S),

which proves the result since CH0(S×S) has no torsion by Roitman’s theorem.

Proof of Theorem 4.2.1. We know that S is swept-out by a 1-parameterfamily of elliptic curves: Thus there is a smooth surface Σ which admits anelliptic fibration

p : Σ → B

and a generically finite dominating morphism

q : Σ → S.

In fact, we can even assume that all fibers Σb are reduced irreducible, if S hasthe property that Pic S is generated by the class of an ample line bundle L onS. In that case, there is a one parameter family of elliptic curves in |L| and allof them are irreducible and reduced. Once we have the result for S satisfyingthis property, the result for any S follows by specialization.

We set d := deg q∗L|Σb. We apply Proposition 4.2.2, (ii) to the elliptic

curve Σb endowed with the line bundle Lb. This gives us for the general pointb ∈ B a formula for the small diagonal of Σb in CH1(Σb ×Σb ×Σb)Q. We viewΣb × Σb × Σb as the fiber of the map

p3 : Σ×B Σ×B Σ → B,

and observe that the small diagonal ∆Σ of Σ is contained in Σ ×B Σ ×B Σand that its restriction to the general fiber Σb × Σb × Σb is equal to the smalldiagonal of Σb. We thus conclude from Corollary 2.1.5 that there are finitelymany points bi ∈ B and 2-cycles Zi ⊂ Σbi × Σbi × Σbi with Q-coefficients suchthat the following equation holds in CH2(Σ×B Σ×B Σ)Q:

d2∆Σ = [d[∆rel12 · pr∗3(q∗L) + (perm.)] (4.2.9)

−(pr∗1q∗L · pr∗2q∗L + (perm.)) +∑

i

Zi,

where ∆rel12 is defined as the inverse image pr−1

12 (∆Σ/B) ⊂ Σ×B Σ×B Σ of therelative diagonal of Σ over B.


We observe furthermore the following: Up to changing the base B, we mayassume there is an involution ι : Σ → Σ, with the properties that

ι∗(q∗L) = q∗L, p ι = p

acting on each smooth fiber Σb in such a way that Σb/ι ∼= P1. Then the cycle

d2∆Σ−[d[∆rel12 ·pr∗3(q∗L)+(perm.)]+(pr∗1q∗L·pr∗2q∗L+(perm.)) ∈ CH2(Σ×BΣ×BΣ)Q

being invariant under ι, we can assume by averaging that each cycle Zi is in-variant under ι.

We now push forward this equality to S × S × S via the composition of theinclusion

k : Σ×B Σ×B Σ → Σ× Σ× Σ

and the map(q, q, q) : Σ× Σ× Σ → S × S × S.

Let N := deg q. Then we clearly have

(q, q, q)∗∆Σ = N∆. (4.2.10)

Next we have the following lemma:

Lemma 4.2.4. The following equality holds in CH2(S × S × S)Q:

((q, q, q) k)∗(pr∗i q∗L · pr∗j q∗L)(4.2.11)

= Nd2pr∗ijo× o + Ndpr∗i o× pr∗j L× pr∗l L + Nd pr∗j o× pr∗i L× pr∗l L,

where i, j, l = 1, 2, 3. Furthermore

((q, q, q) k)∗(∆rel12 · pr∗3(q∗L)) = Nd∆12 · pr∗3o (4.2.12)

+Npr∗1L× o2 × pr∗3L + Npr∗1o · p∗2L · pr∗3L in CH2(S × S × S)Q.

Proof. We may assume that i = 1, j = 2. Let P = P(H0(S, L)) = Pr. Thecurve B is a curve of degree N in P . The cycle ((q, q, q)k)∗(pr∗1q∗L ·pr∗2q∗L) ∈CH2(S × S × S) is N times the cycle

((q′, q′, q′) k′)∗(pr∗1q∗L · pr∗2q∗L) ∈ CH2(S × S × S)

where B is replaced by a pencil P1 ⊂ P ,

q′ : Σ′ → S, p′ : Σ′ → P1

are obtained by blowing-up S at the base points of the pencil, and k′ is theinclusion of Σ′×P1 Σ′×P1 Σ′ in Σ′×Σ′×Σ′. But the small diagonal of P1×P1×P1

has its class equal to∑

i6=j

π∗i H · π∗j H, H := OP1(1),


where πj : P1 × P1 × P1 → P1 is the j-th projection. Hence the class of Σ′ ×P1Σ′×P1 Σ′ in Σ′×Σ′×Σ′ is equal to

∑i 6=j p′∗i H · p′∗jH where p′l = πl (p′, p′, p′).

Our cycle in (4.2.11) is thus equal to

N(q′, q′, q′)∗[(∑

i 6=j

p′i∗H · p′j∗H) · pr∗1q∗L · pr∗2q∗L]

which by the projection formula is also equal to

Npr∗1L · pr∗2L · ((q′, q′, q′)∗(∑

i 6=j

p′i∗H · p′j∗H)).

As q′∗H = L, this is clearly equal to

Npr∗1L · pr∗2L · (∑

i6=j

pr∗i L · pr∗j L)).

For i, j = 1, 2 we get Nd2o1 × o2 × S using (4.1.1). The two other termsgive for the same reason Ndo1×L×L and NdL× o2×L. This proves (4.2.11).

The formula (4.2.12) is proved in a similar way, using Lemma 4.1.5.

We now deduce from (4.2.9), (4.2.10) and (4.2.11), by applying (q, q, q)∗ k∗the following equality in CH2(S × S × S)Q:

Nd2∆ = d[Nd∆12 · pr∗3o + Npr∗1L · pr∗2o · pr∗3L + Npr∗1o · p∗2L · pr∗3L] (4.2.13)

−[Nd2o× o× S −Ndo× L× L−NdL× o× L + (perm.)] +∑

i

Zi

where the Zi are 2-cycles supported on products of curves Ci × Ci × Ci, wherethe Ci ∈ |L| are elliptic, and the Zi are invariant under the symmetric groupS3 and the action of the involution ι.

We have now, denoting by n : Ci → Ci the normalization of Ci :

Lemma 4.2.5. 2-cycles in Ci×Ci×Ci which are invariant under the symmetricgroup S3 and the action of the involution ι are generated over Q by

∑j(n

prj)∗L|Ciand by the big diagonal

∑k,l pr∗kl∆Ci

.

This is elementary as Ci is either elliptic or rational, and n∗(L|Ci) generates

over Q the invariant part under ι of Pic Ci.Using (4.1.1), Lemma 4.1.5 and the above lemma, we conclude that the

cycles Zi are rationally equivalent in S × S × S to pr∗1L · pr∗2o · pr∗3L + (perm).We thus deduce from (4.2.13) an equality

∆− (∆12 · pr∗3o+(perm))+ o× o×S +(perm) = µ(pr∗1L · pr∗2o · pr∗3L+(perm))

in CH2(S × S × S)Q, where µ ∈ Q.Comparing cohomology classes, we find that µ = 0 which concludes the proof

of Theorem 4.2.1.


4.2.1 Calabi-Yau hypersurfaces

In the case of smooth Calabi-Yau hypersurfaces X in projective space Pn, that ishypersurfaces of degree n+1 in Pn, we have the following result which partiallygeneralizes Theorem 4.2.1 and provides some information on the Chow ring ofX. Denote by o ∈ CH0(X)Q the class of the zero cycle hn−1

n+1 , where h :=c1(OX(1)) ∈ CH1(X). We denote again by ∆ the small diagonal of X in X3.

Theorem 4.2.6. (Voisin 2011) The following relation is satisfied in the groupCH2n−2(X ×X ×X)Q:

∆ = ∆12 × o3 + (perm.) + Z + Γ′, (4.2.14)

where Z is the restriction to X ×X ×X of a cycle on Pn × Pn × Pn, and Γ′ isa multiple of the following effective cycle of dimension n− 1:

Γ := ∪l∈F (X)P1l × P1

l × P1l .

Here F (X) is the variety of lines contained in X. It is of dimension n − 4for general X. For a point l ∈ F (X), P1

l ⊂ X denotes the corresponding line.

Proof of Theorem 4.2.6. Observe first of all that it suffices to prove thefollowing equality of n− 1-cycles on X3

0 where X30 := X3 \∆:

Γ|X30

= 2(n + 1)[∆12|X30× o3 + (perm.)] + Z in CH2n−2(X3

0 )Q (4.2.15)

where Z is the restriction to X30 of a cycle on (Pn)3. Indeed, by the localization

exact sequence (1.1.2), (4.2.15) implies an equality

N∆ = ∆12 × o3 + (perm.) + Z + Γ′ in CH2n−2(X ×X ×X)Q, (4.2.16)

for some integer N . Projecting to X2 and taking cohomology classes, we easilyconclude then that N = 1. (We use here the fact that X has some transcendentalcohomology, so that the cohomology class of the diagonal of X does not vanishon products U × U , where U ⊂ X is Zariski open.)

In order to prove (4.2.15), we do the following: First of all we compute theclass in CHn−1(X3

0 ) of the 2n− 2-dimensional subvariety

X30,col,sch ⊂ X3

0

parameterizing 3-uples of colinear points satisfying the following property:Let P1

x1x2x3:=< x1, x2, x3 > be the line generated by the xi’s. Then the

subscheme x1 + x2 + x3 of P1x1x2x3

⊂ Pn is contained in X.We will denote

X30,col ⊂ X3

0

the 2n − 2-dimensional subvariety parameterizing 3-uples of colinear points.Obviously X3

0,col,sch ⊂ X30,col. We will see that it is in fact one irreducible

component of it.


Next we observe that there is a natural morphism φ : X30,col → G(2, n + 1)

to the Grassmannian of lines in Pn, which to (x1, x2, x3) associates the lineP1

x1x2x3. This morphism is well-defined on X3

0,col because at least two of thepoints xi are distinct, so that this line is unique. The morphism φ correspondsto a tautological rank 2 vector bundle E on X3

0,col, with fiber H0(OP1x1x2x3(1))

over the point (x1, x2, x3).We then observe that Γ ⊂ X3

0,col,sch is defined by the condition that theline P1

x1x2x3is contained in X. In other words, the equation f defining X has

to vanish on the line P1x1x2x3

. This equation can be seen globally as giving asection σ,

σ((x1, x2, x3)) = f|P1x1x2x3

of the vector bundle Sn+1E .This section σ is not transverse, (in fact the rank of Sn+1E is n + 2, while

the codimension of Γ is n − 1), but the reason for this is very simple: indeed,at a point (x1, x2, x3) of X3

0,col,sch, the equation f vanishes by definition on thedegree 3 cycle x1 + x2 + x3 of P1

x1x2x3. Another way to express this is to say

that σ is in fact a section of the rank n− 1 bundle

F ⊂ Sn+1E (4.2.17)

where F(x1,x2,x3) consists of degree n+1 polynomials vanishing on the subschemex1 + x2 + x3 of P1

x1x2x3.

The section σ of F is transverse and thus we conclude that we have thefollowing equality

Γ|X30

= j∗(cn−1(F)) in CH2n−2(X30 )Q, (4.2.18)

where j is the inclusion of X30,col,sch in X3

0 .We now observe that the vector bundles E and F come from vector bundles

on the variety (Pn)30,col parameterizing 3-uples of colinear points in Pn, at leasttwo of them being distinct.

The variety (Pn)30,col is smooth irreducible of dimension 2n + 1 (hence ofcodimension n− 1 in (Pn)3), being Zariski open in a P1 × P1 × P1-bundle overthe Grassmannian G(2, n + 1). We have now the following:

Lemma 4.2.7. The intersection (Pn)30,col ∩ X30 is reduced, of pure dimension

2n− 2. It decomposes as

(Pn)30,col ∩X30 = X3

0,col,sch ∪∆0,12 ∪∆0,13 ∪∆0,23, (4.2.19)

where ∆0,ij ⊂ X30 is defined as ∆ij \∆ with ∆ij the big diagonal xi = xj.

Proof. The set theoretic equality in (4.2.19) is obvious. The fact that eachcomponent on the right has dimension 2n − 2 and thus is a component of theright dimension of this intersection is also obvious. The only point to checkis thus the fact that these intersections are transverse at the generic point of


each component in the right hand side. The generic point of the irreduciblevariety X3

0,col,sch parameterizes a triple of distinct colinear points which areon a line ∆ not tangent to X. At such a triple, the intersection (Pn)30,col ∩X3

0 is smooth of dimension 2n − 2 because (Pn)30,col is the triple self-productP×G(2,n+1)P×G(2,n+1)P of the tautological P1-bundle P over the GrassmannianG(2, n + 1), and the intersection with X3

0 is defined by the three equations

p pr∗1f, p pr∗2f, p pr∗3f,

where the pri’s are the projections P 3/G(2,n+1) → P and p : P → Pn is thenatural map. These three equations are independent since they are independentafter restriction to P1

x1x2x3×P1

x1x2x3×P1

x1x2x3⊂ (Pn)30,col at the point (x1, x2, x3)

because P1x1x2x3

is not tangent to X.Similarly, the generic point of the irreducible variety ∆0,12 ⊂ X3

0,col parame-terizes a triple (x, x, y) with the property that x 6= y and the line P1

xy :=< x, y >is not tangent to X. Again, the intersection (Pn)30,col ∩X3

0 is smooth of dimen-sion 2n − 2 near (x, x, y) because the restrictions to P1

xy × P1xy × P1

xy of theequations

p pr∗1f, p pr∗2f, p pr∗3f,

defining X3 are independent.

Combining (4.2.19), (4.2.18) and the fact that the vector bundle F alreadyexists on (Pn)30,col, we find that

Γ|X30

= J∗(cn−1(F|(Pn)30,col∩X30))−

∑

i 6=j

Jij∗cn−1(F|∆0,ij) in CH2n−2(X3

0 )Q,

where J : (Pn)30,col ∩X30 → X3

0 is the inclusion and similarly for J0ij : ∆0,ij →X3

0 . This provides us with the formula:

Γ|X30

= (K∗cn−1(F))|X30−

∑

i 6=j

Jij∗cn−1(F|∆0,ij) in CH2n−2(X3

0 )Q, (4.2.20)

where K : (Pn)30,col → (Pn)30 is the inclusion map.The first term comes from CH((Pn)30), and this only contributes to the

term Z in Theorem 4.2.6, so to conclude we only have to compute the termsJij∗cn−1(F|∆0,ij

). This is however very easy, because the vector bundles E andF are very simple on ∆0,ij : Assume for simplicity i = 1, j = 2. Points of ∆0,ij

are points (x, x, y), x 6= y ∈ X. The line φ((x, x, y)) is the line P1xy =< x, y >,

and it follows that

E|∆0,12 = pr∗2OX(1)⊕ pr∗3OX(1). (4.2.21)

The projective bundle P(E|∆0,12) has two sections on ∆12 which give two divisors

D2 ∈ |OP(E)(1)⊗ pr∗3OX(−1)|, D3 ∈ |OP(E)(1)⊗ pr∗2OX(−1)|.


The length 3 subscheme 2D2 + D3 ⊂ P(E|∆0,1,2) with fiber 2x + y over thepoint (x, x, y) is thus the zero set of a section α of the line bundle OP(E)(3) ⊗pr∗3OX(−2) ⊗ pr∗2OX(−1). We thus conclude that the vector bundle F|∆0,12 isisomorphic to

pr∗3OX(2)⊗ pr∗2OX(1)⊗ Sn−2E|∆0,12 .

Combining with (4.2.21), we conclude that cn−1(F|∆0,ij) can be expressed as a

polynomial of degree n − 1 in h2 = c1(pr∗2OX(1)) and h3 = c1(pr∗3OX(1))) on∆0,12. The proof of (4.2.15) is completed by the following lemma:

Lemma 4.2.8. Let ∆X ⊂ X × X be the diagonal. Then the codimension ncycles

pr∗1c1(OX(1)) ·∆X , pr∗2c1(OX(1)) ·∆X

of X ×X are restrictions to X ×X of cycles in CHn(Pn × Pn)Q.

Proof. Indeed, let jX : X → Pn be the inclusion of X in Pn, and jX,1, jX,2

the corresponding inclusions of X ×X in Pn,×X, resp. X × Pn. Then as X isa degree n + 1 hypersurface, we have

(n + 1)pr∗1c1(OX(1))· = j∗X,1 jX,1∗ : CH(X ×X) → CH(X ×X),

and similarly for the second inclusion. On the other hand, jX,1∗(∆X) ⊂ Pn×Xis obviously the (transpose of the) graph of the inclusion of X in Pn, hence itsclass is the restriction to Pn×X of the diagonal of Pn×Pn. We argue similarlyfor the second inclusion.

It follows from this lemma that a monomial of degree n−1 in h2 = c1(pr∗2OX(1))and h3 = c1(pr∗3OX(1))) on ∆0,12, seen as a cycle in X × X × X, will be therestriction to X × X × X of a cycle with Q-coefficients, unless it is propor-tional to hn−1

3 . Recalling that c1(OX(1))n−1 = (n + 1)o ∈ CH0(X), we finallyproved that modulo restrictions of cycles coming from Pn × Pn × Pn, the termJ12∗cn−1(F|∆0,12) is a multiple of ∆12×o3 in CH2n−2(X×X×X)Q). The precisecoefficient is in fact given by the argument above. Indeed, we just saw that mod-ulo restrictions of cycles coming from Pn×Pn×Pn, the term J12∗cn−1(F|∆0,12)is equal to

µ∆12 · pr∗3(c1(OX(1))n−1) = µ(n + 1)∆12 × o3, (4.2.22)

with c1(OX(1))n−1 = (n + 1)o in CH0(X), and where the coefficient µ is thecoefficient of hn−1

3 in the polynomial in h2, h3 computing cn−1(F|∆0,12).We use now the isomorphism

F|∆0,12∼= pr∗3OX(2)⊗ pr∗2OX(1)⊗ Sn−2E|∆0,12 ,

where E|∆0,12∼= pr∗2OX(1)⊕pr∗3OX(1) according to (4.2.21). Hence we conclude

that the coefficient µ is equal to 2, and this concludes the proof of (4.2.15), using(4.2.22) and (4.2.20).

4.3. DELIGNE’S DECOMPOSITION THEOREM 101

We have the following consequence of Theorem 4.2.6, which is a generaliza-tion of Theorem 4.1.1 to Calabi-Yau hypersurfaces.

Theorem 4.2.9. Let Zi, Z ′i, i = 1, . . . , N be cycles on X such that

codim Zi > 0, ∀i, codim Zi + codim Z ′i = n− 1.

Then if we have a cohomological relation∑

i

ni[Zi] ∪ [Z ′i] = 0 in H2n−2(X,Q)

this relation already holds at the level of Chow groups:∑

i

niZi · Z ′i = 0 in CHn−1(X)Q.

Proof. Indeed, let us view formula (4.2.14) as an equality of correspondencesbetween X ×X and X. The left hand side applied to

∑i niZi × Z ′i is

∆∗(∑

i

niZi × Z ′i) =∑

i

niZi · Z ′i in CH(X)Q.

The right hand side is a sum Z∗(∑

i niZi × Z ′i) + Γ∗(∑

i niZi × Z ′i). But weobserve that as Z is the restriction of a cycle Z ′ ∈ CH2n−2(Pn × Pn × Pn)Q,Z∗(

∑i niZi × Z ′i) is equal to

j∗(Z ′∗((j, j)∗(∑

i

niZi × Z ′i))) ∈ CHn−1(X)Q.

As j∗(Z ′∗((j, j)∗(∑

i niZi×Z ′i))) is cohomologous to 0 on X, Z ′∗((j, j)∗(∑

i niZi×Z ′i)) is cohomologous to 0 on Pn. Hence it is rationally equivalent to 0 on Pn

and so is j∗(Z ′∗((j, j)∗(∑

i niZi ×Z ′i))) = Z∗(∑

i niZi ×Z ′i). Consider now theterm Γ∗(

∑i niZi×Z ′i): Let Γ0 ⊂ X be the locus swept-out by lines. We observe

that for any line ∆ ⊂ X, any point on ∆ is rationally equivalent to the zerocycle h ·∆ which is in fact proportional to o, since

(n + 1)h ·∆ = j∗ j∗(∆) in CH(X)

and j∗(∆) = c1(OPn(1))n−1) in CHn−1(Pn). Hence all points of Γ0 are ratio-nally equivalent in X. It follows that the 0-cycle Γ∗(

∑i niZi × Z ′i), which is of

degree 0 and supported on Γ0 is in fact rationally equivalent to 0.

4.3 Deligne’s decomposition theorem for fami-lies of K3 surfaces

4.3.1 Deligne’s decomposition theorem

Let φ : X → Y be a submersive and proper morphism of complex varieties.Recall that the morphism φ is projective if there exists a holomorphic embedding

i : X → Y × Pn


such that φ = pr1 i. Giving such a embedding provides a class ω ∈ H2(X,Z)defined by

ω = (pr2 i)∗c1(OPn(1)).

As pr2 i|Xtis a holomorphic immersion on each fiber Xt of φ, the restriction

ωt := ω|Xt∈ H2(Xt,Z)

is a Kahler class, and the morphism

ωt∪ : Hk(Xt,Q) → Hk+2(Xt,Q)

is a Lefschetz operator on H∗(Xt,Q). Moreover, as ω is closed, it induces asabove a morphism of local systems

L := ω∪ : R∗φ∗Q→ R∗+2φ∗Q,

which is equal to Lt = ωt∪ on the fiber at the point t. The operator L is calledthe relative Lefschetz operator. If n = dim Xt is the relative dimension of φ, weknow that Lt satisfies the hard Lefschetz theorem, i.e.

Ln−kt : Hk(Xt,Q) → H2n−k(Xt,Q) (4.3.23)

is an isomorphism for k ≤ n.One deduces formally from the Lefschetz isomorphisms the Lefschetz decom-

position

Hk(Xt,Q) = ⊕2r≤kLrHk−2r(Xt,Q)prim for k ≤ n, (4.3.24)

where

Hk−2r(Xt,Z)prim := Ker Ln−k+2r+1 : Hk−2r(Xt,Q)prim → H2n−k+2r+2(Xt,Q).

The corresponding decomposition for k ≥ n is obtained by applying the isomor-phism (4.3.23).

The relative Lefschetz operator thus gives relative Lefschetz isomorphisms

Ln−k : Rkφ∗Q ∼= R2n−kφ∗Q

and a relative Lefschetz decomposition

Rkφ∗Q = ⊕2r≤kLrRk−2rφ∗Qprim, k ≤ n.

The relative Lefschetz operator L as well as its powers Lk induce endomor-phisms (of degree 2k) of the Leray spectral sequence of φ, i.e. of the morphismsLk

r of the complexes (Ep,qr , dr) (of degree 2k on the second index), and the mor-

phism Lk is compatible with the Leray filtration and with each Lk∞ on H∗(X,Q).

Finally, the Lefschetz isomorphism (4.3.23) shows that

Ln−k2 : H l(Y, Rkφ∗Q) → H l(Y, R2n−kφ∗Q) (4.3.25)

is an isomorphism for k ≤ n.Let us recall the proof of the following theorem, due to Deligne [28].


Theorem 4.3.1. If φ : X → Y is a submersive projective morphism, the Lerayspectral sequence of φ degenerates at E2.

Proof. Let us first show that d2 = 0. For this, note that if q ≥ n, we havethe following commutative diagramme:

Ep,2n−q2 = Hp(Y, R2n−qφ∗Q)

Lq−n2 //

d2

²²

Ep,q2 = Hp(Y, Rqφ∗Q)

d2

²²Ep+2,2n−q−1

2 = Hp+2(Y, R2n−q+1φ∗Q) // Ep+2,q−12 = Hp+2(Y, Rq−1φ∗Q),

where the upper horizontal arrow is an isomorphism. It thus suffices to showthat d2 = 0 on Ep,q

2 with q ≤ n. We have the decomposition

Ep,q2 = ⊕2r≤qL

r2H

p(Y,Rq−2rφ∗Qprim)

induced by the relative Lefschetz decomposition, and it suffices to show thatd2 = 0 on Lr

2Hp(Y, Rq−2rφ∗Qprim). As Lr

2 commutes with d2, it suffices toshow that d2 = 0 on Hp(Y, Rq−2rφ∗Qprim) ⊂ Ep,q−2r

2 .Setting k = q − 2r, we have the following commutative diagramme:

Ln−k+12 : Hp(Y,Rkφ∗Qprim) //

d2

²²

Hp(Y, R2n−k+2φ∗Q)

d2

²²Ln−k+1

2 : Hp+2(Y, Rk−1φ∗Qprim) // Hp+2(Y, R2n−k+1φ∗Q).

The upper arrow is 0 by definition of the primitive cohomology, while the lowerarrow is the isomorphism (4.3.25). Thus, the first arrow d2 is zero.

To show that the arrows dr, r ≥ 2 are also zero, we proceed in exactly thesame way, using the morphisms of spectral sequences Lk

r and noting that ifds = 0 for 2 ≤ s < r, then Ep,q

r = Ep,q2 , so that we can use the Lefschetz

decomposition as above on Ep,qr .

As explained in [28], the proof above has the following much stronger con-sequence:

Theorem 4.3.2. (Deligne 1968) In the derived category of sheaves of Q-vectorspaces on B, there is a decomposition

Rπ∗Q = ⊕iRiπ∗Q[−i]. (4.3.26)

Proof. We simply observe that the arguments given for the degeneracy atE2 of the Leray spectral sequence of π relative to the constant sheaf Q or Rprove as well the degeneracy at E2 of the Leray spectral sequence of π relativeto the locally constant sheaves π−1((Rkπ∗Q)∗).

We deduce from this that the natural map

Hk(X, π−1((Rkπ∗Q)∗)) → H0(B,Rkπ∗(π−1((Rkπ∗Q)∗)))


is surjective. The right hand side is equal to H0(B, Rkπ∗Q⊗(Rkπ∗Q)∗) and thuscontains the identity βk of Rkπ∗Q. There is thus a class αk ∈ Hk(X, π−1((Rkπ∗Q)∗))which induces βk.

We view αk as giving a morphism

Rkπ∗Q[−k] → Rπ∗Q.

This morphism by definition induces the identity on cohomology of degree kand 0 otherwise. The direct sum of the morphisms αk thus provides the desiredquasiisomorphism.

4.3.2 Multiplicative decomposition isomorphisms

Note that both sides of (4.3.26) carry a cup-product. On the right, we put thedirect sum of the relative cup-product maps µi,j : Riπ∗Q⊗Rjπ∗Q→ Ri+jπ∗Q.On the left, one needs to choose an explicit representation of Rπ∗Q by a complexC∗, together with an explicit morphism of complexes µ : C∗ ⊗ C∗ → C∗ whichinduces the cup-product in cohomology. When passing to coefficients R or C, onecan take C∗ = π∗A∗X , where A∗X is the sheaf of C∞ real or complex differentialforms on X and for µ the wedge product of forms. For rational coefficients, theexplicit construction of the cup-product at the level of complexes (for exampleCech complexes) is more painful (see [37, 6.3]). The resulting cup-productmorphism µ will be canonical only in the derived category. The question westudy in the last two sections is the following:

Question 4.3.3. Given a family of smooth projective varieties π : X → B,does there exist a decomposition as in (4.3.26) which is multiplicative, that iscompatible with the morphism

µ : Rπ∗Q⊗Rπ∗Q→ Rπ∗Q

given by cup-product?

In fact, we will rather consider the following variant: For which class of va-rieties X does there exist a multiplicative decomposition isomorphism as abovefor any family of deformations of X ? The simplest example is that of projectivebundles π : P(E) → B, where E is a locally free sheaf on B.

Lemma 4.3.4. (cf. [95]) Assume that ctop1 (E) = 0 in H2(B,Q). Then, if

there exists a multiplicative decomposition isomorphism, one has ctopi (E) = 0 in

H2i(B,Q) for all i > 0.

Proof. Let h = ctop1 (OP(E)(1)) ∈ H2(P(E),Q). It is standard that

H2(P(E),Q) = π∗H2(B,Q)⊕Qh,

where π∗H2(B,Q) identifies canonically with the deepest term H2(B, R0π∗Q)in the Leray spectral sequence. A multiplicative decomposition isomorphism as


in (4.3.26) induces by taking cohomology another decomposition of H2(P(E),Q)as π∗H2(B,Q) ⊕ Qh′, where h′ = h + π∗α, for some α ∈ H2(B,Q). Inthis multiplicative decomposition, h′ will generate a summand isomorphic toH0(B,R2π∗Q). Let r = rank E . As ctop

1 (E) = 0, one has π∗hr = 0 inH2(B,Q). As (h′)r = 0 in H0(B, R2rπ∗Q), and (h′)r belongs by multiplica-tivity to a direct summand naturally isomorphic (by restriction to fibers) toH0(B,R2rπ∗Q) = 0, one must also have (h′)r = 0 in H2r(P(E),Q). On theother hand (h′)r = hr + rhr−1π∗α + . . . + π∗αr, and it follows that

π∗(h′)r = 0 = π∗hr + rα in H2(B,Q).

Thus α = 0, h′ = h, and hr = 0 in H2r(P(E),Q). The definition of Chernclasses and the fact that hr = 0 shows then that ctop

i (E) = 0 for all i > 0.

In this example, the obstructions to the existence of a multiplicative de-composition isomorphism are algebraic classes on the base B. They vanish ondense Zariski open sets of B, and this suggests studying the following variantof Question 4.3.3:

Question 4.3.5. Given a family of smooth projective varieties π : X → B, doesthere exist a Zariski dense open set B0 of B, and a multiplicative decompositionisomorphism as in (4.3.26) for the restricted family X 0 → B0?

We will give a simple example where Question 4.3.5 has a negative answer.It is based on the following criterion (Proposition 4.3.6): Let π : X → B bea projective family of smooth complex varieties without irregularity, parame-terized by a complex quasi-projective variety B. Let Li, i = 1, . . . ,m be linebundles on X and li := ctop

1 (Li) ∈ H2(X ,Q).

Proposition 4.3.6. Assume that there is a multiplicative decomposition iso-morphism

Rπ∗Q = ⊕iRiπ∗Q[−i].

Then for any fiberwise cohomological relation

P (li,b) = 0 in H2r(Xb,Q) ∀b ∈ B,

where P is a homogeneous polynomial of degree r in m variables with rationalcoefficients, the class P (li) ∈ H2r(X ,Q) vanishes locally over B in the Zariskitopology, that is, B is covered by Zariski open sets B0 ⊂ B, such that

P (li)|X 0 = 0 in H2r(X 0,Q),

where X 0 = π−1(B0).

Proof. We will assume for simplicity that B is smooth although a closerlook at the proof shows that this assumption is not necessary. The multiplicativedecomposition isomorphism induces, by taking cohomology and using the factthat the fibers have no degree 1 rational cohomology, a decomposition

H2(X ,Q) = H0(B, R2π∗Q)⊕ π∗H2(B,Q), (4.3.27)


which is compatible with cup-product, so that the cup-product map on the firstterm factors through the map induced by cup-product:

µr : H0(B,R2π∗Q)⊗r → H0(B,R2rπ∗Q).

We write in this decomposition li = l′i + π∗ki, where ki ∈ H2(B, R0π∗Q) =

H2(B,Q)π∗∼= π∗H2(B,Q). We claim that the ki are divisor classes on B. Indeed,

take any line bundle L on X . Let l = c1top(L) ∈ H2(X ,Q) and decompose asabove l = l′ + π∗k, where l′ has the same image as l in H0(B, R2π∗Q) and kbelongs to H2(B,Q). Denoting by n the dimension of the fibers, we get:

lnli = (∑

p

(n

p

)l′pπ∗kn−p)(l′i + π∗ki) (4.3.28)

=∑

p

(n

p

)l′pπ∗kn−pl′i +

∑p

(n

p

)l′pπ∗kn−pπ∗ki.

Recall now that the decomposition is multiplicative. The class l′nl′i thus belongsto a direct summand of H2n+2(X ,Q) isomorphic to H0(B, R2n+2π∗Q) = 0.Hence it follows that it is identically 0. Applying π∗ to (4.3.28), we then get:

π∗(lnli) = degXb(l′n)ki + ndegXb

(l′n−1l′i)k (4.3.29)

= degXb(ln)ki + ndegXb

(ln−1li)k.

Observe that the term on the left is a divisor class on B. If the fiberwise self-intersection degXb

(lin) is non zero, we can take L = Li and (4.3.29) shows thatki is a divisor class on B as claimed. If it is 0, choose a line bundle L on Ssuch that the fiberwise intersection numbers degXb

(ln−1li) and degXb(ln) do not

vanish (such an L exists because the morphism π is projective). Then, applying(4.3.29) to both pairs (L,L) and (L,Li) shows that both k and degXb

(ln)ki +ndegXb

(ln−1li)k are divisor classes on B, so that ki is also a divisor class on B.As divisor classes are locally trivial on B for the Zariski topology, we thus

have that, locally on B for the Zariski topology, ki = 0 and thus li belongs to thefirst summand H0(B, R2π∗Q) in (4.3.27). Let B0 be a Zariski open set wherethis is the case and let X 0 = π−1(B0). It then follows by multiplicativity thatany polynomial expression P (li)|X 0 belongs to a direct summand of H2r(X 0,Q)isomorphic by the natural projection to H0(B0, R2rπ∗Q).

Consider now our fiberwise cohomological polynomial relation P (li,b) = 0 inH2r(Xb,Q), for b ∈ B. It says equivalently that P (li) vanishes in H0(B0, R2rπ∗Q).It follows then from the previous statement that it vanishes in H2r(X 0,Q).

We consider now a smooth projective surface S, and set

X = S × S∆, B = S, π = pr2 τ,

where τ : S × S∆ → S is the blow-up of the diagonal.


Proposition 4.3.7. Assume that h1,0(S) = 0, h2,0(S) 6= 0. Then there for themorphism π : X → B above, there is no multiplicative decomposition isomor-phism over any Zariski dense open set of B = S.

Proof. Let H be an ample line bundle on S, and d := deg c1(H)2. On X,we have then two line bundles, namely L := τ∗(pr∗1H) and L′ = OX(E) whereE is the exceptional divisor of τ . On the fibers of π, we have the relation

deg c1(L)2 = −ddeg c1(E)2.

If there existed a multiplicative decomposition isomorphism over a Zariski denseopen set of B = S, we would have by Proposition 4.3.6, using the fact that thefibers of π are regular surfaces, a Zariski dense open set U ⊂ S and the relation

ctop1 (L)2 = −dctop

1 (E)2 (4.3.30)

in H4(XU ,Q). If we apply τ∗ to this relation, we now get:

pr∗1ctop1 (H)2 = −d[∆] (4.3.31)

in H4(S × U,Q).This relation implies that the class pr∗1ctop

1 (H)2+d[∆] ∈ H4(S×S,Q) comesfrom a class γ ∈ H2(S × D,Q), where D := S \ U and D is a desingularizationof D. Denoting by j : D → S the natural map, we then conclude that for anyclass α ∈ H2(S,Q),

dα ∈ H2(S,Q) = j∗(γ∗α)

is supported on D. This contradicts the assumption h2,0(S) 6= 0.

4.3.3 Families of abelian varieties

In the abelian case, the existence of a multiplicative decomposition isomorphismis essentially due to Deninger and Murre [30]. Our proof will be based on thefollowing lemma, applied to the category of sheaves of Q-vector spaces on B.

Let A be an abelian category in which morphisms are Q-vector spaces, andlet D(A) be the corresponding derived category of left bounded complexes. LetM ∈ D(A) be an object with bounded cohomology such that End M is finitedimensional. Assume M admits a morphism φ : M → M such that

Hi(φ) : Hi(M) → Hi(M)

is equal to λiIdHi(M), where all the λi ∈ Q are distinct.

Lemma 4.3.8. The morphism φ induces a canonical decomposition

M ∼= ⊕iHi(M)[−i], (4.3.32)

characterized by the properties :1) The induced map on cohomology is the identity map.


2) One has

φ πi = λiπi : M → M. (4.3.33)

where πi corresponds via the isomorphism (4.3.32) to the i-th projector pri.

Proof. We first prove using the arguments of [28] that M is decomposed,namely there is an isomorphism

f : M ∼= ⊕iHi(M)[−i].

For this, given an object K ∈ Ob A, we consider the left exact functor T fromA to the category of Q-vector spaces defined by T (N) = HomA(K,N), and forany integer i the induced functor, denoted by Ti, N 7→ HomD(A)(K[−i], N) onDb(A). For any N ∈ Db(A), there is the hypercohomology spectral sequencewith E2-term

Ep,q2 = RpTi(Hq(N)) = Extp+i

A (K, Hq(N)) ⇒ Rp+qTi(N).

Under our assumptions, this spectral sequence for N = M degenerates at E2.Indeed, the morphism φ acts then on the above spectral sequence starting fromE2. The differential d2 : Ep,q

2 → Ep+2,q−12

Extp+iA (K, Hq(M)) ⇒ Extp+2+i

A (K, Hq−1(M)) (4.3.34)

commutes with the action of φ. On the other hand, φ acts as λqId on the lefthand side and as λq−1Id on the right hand side of (4.3.34). Thus we concludethat d2 = 0 and similarly that all dr, r ≥ 2 are 0.

We take now K = Hi(M). We conclude from the degeneracy at E2 of theabove spectral sequence that the map

HomD(A) (Hi(M)[−i],M) → HomA(Hi(M),Hi(M)) = E−i,i2

is surjective, so that there is a morphism

fi : Hi(M)[−i] → M

inducing the identity on degree i cohomology. The direct sum f =∑

fi is aquasi-isomorphism which gives the desired splitting.

The morphism φ can thus be seen as a morphism of the split object⊕iHi(M)[−i].

Such a morphism is given by a block-uppertriangular matrix

φj,i ∈ Exti−jA (Hi(M),Hj(M)), i ≥ j,

with λiId on the i-th diagonal block. Let ψ be the endomorphism of End Mgiven by left multiplication by φ. We have by the above description of φ:

∏

i,Hi(M)6=0

(ψ − λiIdEnd M ) = 0, (4.3.35)


which shows that the endomorphism ψ is diagonalizable. More precisely, as ψis block-uppertriangular in an adequately ordered decomposition

End M = ⊕i≥jExti−jA (Hi(M), Hj(M)),

with term λjId on the block diagonals Exti−jA (Hi(M),Hj(M)), hence in par-

ticular on EndA Hj(M), we conclude that there exists π′i ∈ End M such that π′iacts as the identity on Hi(M), and φ π′i = λiπ

′i.

Let ρi := π′i fi : Hi(M)[−i] → M . Then ρ :=∑

ρi gives another decompo-sition ⊕iH

i(M)[−i] ∼= M and we have φ ρi = λiρi, which gives φ πi = λiπi,where πi = ρ pri ρ−1.

The uniqueness of the πi satisfying properties 1) and 2) is obvious, sincethese properties force the equality πi =

∏j 6=i(φ−λjIdM )∏

j 6=i λi−λj.

Corollary 4.3.9. For any family π : A → B of abelian varieties (or complextori), there is a multiplicative decomposition isomorphism Rπ∗Q = ⊕iR

iπ∗Q[−i].

Proof. Choose an integer n 6= ±1 and consider the multiplication map

µn : A → A, a 7→ na.

We then get morphisms µ∗n : Rπ∗Q→ Rπ∗Q with the property that the inducedmorphisms on each Riπ∗Q = Hi(Rπ∗Q) is multiplication by ni. We use nowLemma 4.3.8 to deduce from such a morphism a canonical splitting

Rπ∗Q ∼= ⊕iRiπ∗Q[−i], (4.3.36)

characterized by the properties that the induced map on cohomology is theidentity map, and

µ∗n πi = niπi : Rπ∗Q→ Rπ∗Q. (4.3.37)

where πi is the endomorphism of Rπ∗Q which identifies to the i-th projector pri

via the isomorphism (4.3.36). On the other hand, the morphism µ : Rπ∗Q ⊗Rπ∗Q→ Rπ∗Q given by cup-product is compatible with µ∗n, in the sense that

µ (µ∗n ⊗ µ∗n) = µ∗n µ : Rπ∗Q⊗Rπ∗Q→ Rπ∗Q.

Combining this last equation with (4.3.37), we find that

µ (µ∗n ⊗ µ∗n) (πi ⊗ πj) = ni+jµ (πi ⊗ πj)

= µ∗n µ (πi ⊗ πj) : Rπ∗Q⊗Rπ∗Q→ Rπ∗Q

from which it follows applying again (4.3.37) that µ πi ⊗ πj factors throughRi+jπ∗[−i − j], or equivalently that in the splitting (4.3.36), the cup-productmorphism µ maps Riπ∗Q[−i]⊗Rjπ∗Q[−j]) to the summand Ri+jπ∗[−i− j].


4.3.4 A multiplicative decomposition theorem for familiesof K3 surfaces

The following is one of the main results of [95]. It provides an unexpectedapplication of Theorem 4.1.3.

Theorem 4.3.10. i) For any smooth projective family π : X → B of K3 sur-faces, there exist a nonempty Zariski open subset B0 of B, and a multiplicativedecomposition isomorphism as in (4.3.26) for the restricted family π : X 0 → B0.

ii) Furthermore, the class of the diagonal [∆X0/B0 ] ∈ H4(X ×B X,Q) be-longs to the direct summand H0(B0, R4(π, π)∗Q) of H4(X0×B0 X0,Q), for theinduced decomposition of R(π, π)∗Q.

iii) For any algebraic line bundle L on X , there is a dense Zariski open setB0 of B such that the topological Chern class ctop

1 (L) ∈ H2(X ,Q) restricted toX 0 belongs to the direct summand H0(B0, R2π∗Q) of H2(X 0,Q) induced by thisdecomposition.

In the second statement, (π, π) : X0×B0 X0 → B0 denotes the natural map.A decomposition Rπ∗Q ∼= ⊕iR

iπ∗Q[−i] induces a decomposition

R(π, π)∗Q = ⊕iRi(π, π)∗Q[−i]

by the relative Kunneth isomorphism

R(π, π)∗Q ∼= Rπ∗Q⊗Rπ∗Q.

Theorem 4.3.10, i) is definitely wrong if we do not restrict to a Zariski openset (cf. [95] for an example).

Remark 4.3.11. It follows from Proposition 4.3.6 that statement iii) in The-orem 4.3.10 is in fact a consequence of i).

Proof of Theorem 4.3.10. We use the existence of the canonical 0-cycleot ∈ CH0(Xt) (cf. section 4.1). This cycle may not be spread-up on thetotal space X → B, but this can be done on a generically finite proper coverr : B′ → B, thus providing a cycle oX′ ∈ CH2(X ′), where X ′ = X ×B B′, withoX′|X′

t= ot in CH0(X ′

t). The cycle

1deg r

r′∗oX′ =: oX ∈ CH2(X)Q,

where r′ : X ′ = X ×B B′ → X is the first projection, has then the propertythat oX|Xt

= ot in CH0(Xt)Q. In particular it has degree 1.The cohomology classes

pr∗1 [oX ] =: [Z1], pr∗2 [oX ] =: [Z2] ∈ H4(X ×B X,Q)

of the two codimension 2 cycles pr∗1oX and pr∗2oX , where

pri : X ×B X → B


are the two projections, provide morphisms in the derived category:

P1 : Rπ∗Q→ Rπ∗Q, P2 : Rπ∗Q→ Rπ∗Q

P1 := pr2∗ (pr∗1 [oX ]∪) pr∗1 , P2 := pr2∗ (pr∗2 [oX ]∪) pr∗1 . (4.3.38)

Lemma 4.3.12. (i) The morphisms P1, P2 are projectors of Rπ∗Q.(ii) P1 P2 = P2 P1 = 0 over a Zariski dense open set of B.

Proof. (i) We compute P1 P1. From (4.3.38), (1.1.5) and the projectionformula (1.1.4), we get that P1 P1 is the morphism Rπ∗ → Rπ∗ induced bythe following cycle class

p13∗(p∗12[o1X ] ∪ p∗23[o

1X ]) ∈ H4(X ×B X,Q), (4.3.39)

where the pij are the various projections from X ×B X ×B X to X ×B X. Weuse now the fact that p∗12[o

1X ] = p∗1[oX ], p∗23[o

1X ] = p∗2[oX ], where the pi’s are

the various projections from X ×B X ×B X to X, so that (4.3.39) is equal to

p13∗(p∗1[oX ] ∪ p∗2[oX ]). (4.3.40)

Using the projection formula, this class is equal to

pr∗1 [oX ] ∪ pr∗2(π∗[oX ]) = pr∗1 [oX ] ∪ pr∗2(1B) = pr∗1 [oX ].

This completes the proof for P1 and the proof for P2 is exactly similar.(ii) We compute P1 P2 : From (4.3.38) and (1.1.5), we get that P1 P2 is

the morphism Rπ∗ → Rπ∗ induced by the following cycle class

p13∗(p∗12[o2X ] ∪ p∗23[o

1X ]) ∈ H4(X ×B X,Q), (4.3.41)

where the pij are the various projections from X ×B X ×B X to X ×B X. Weuse now the fact that p∗12[o

2X ] = p∗2[oX ], p∗23[o

1X ] = p∗2[oX ], where the pi’s are

the various projections from X ×B X ×B X to X, so that (4.3.41) is equal to

p13∗(p∗2[oX ] ∪ p∗2[oX ]). (4.3.42)

But the class p∗2[oX ]∪ p∗2[oX ] = p∗2([oX · oX ]) vanishes over a Zariski dense openset of B since the cycle oX · oX has codimension 4 in X. This shows thatP1 P2 = 0 over a Zariski dense open set of B and the proof for P2 P1 worksin the same way.

Using Lemma 4.3.12, we get (up to passing to a Zariski dense open set ofB) a third projector

P := Id− P1 − P2

acting on Rπ∗Q and commuting with the two other ones.


It is well-known (cf. [63]) that the action of these three projectors are

P1∗ = 0 on R2π∗Q, R4π∗Q, P1∗ = Id on R0π∗Q.

P2∗ = 0 on R2π∗Q, R0π∗Q, P2∗ = Id on R4π∗Q.

P∗ = Id on R2π∗Q, P∗ = 0 on R0π∗Q, R4π∗Q.

As a consequence, we get (for example using Lemma 4.3.8) a decomposition

Rπ∗Q ∼= ⊕Riπ∗Q[−i], (4.3.43)

where the corresponding projectors π0, π2, π4 of Rπ∗Q identify respectively toP1, P, P2.

We now prove the following result,

Proposition 4.3.13. Assume the cohomology class of the relative small diago-nal ∆ ⊂ X ×B X ×B X satisfies the equality

[∆] = p∗1[oX ] ∪ p∗23[∆X ] + (perm.)− (p∗1[oX ] ∪ p∗2[oX ] + (perm.)) (4.3.44)

where the pij , pi’s are as above and ∆X is the relative diagonal X ⊂ X ×B X,then, over some Zariski dense open set B0 ⊂ B, we have:

(i) The decomposition (4.3.43) is multiplicative.(ii) The class of the diagonal [∆X ] ∈ H4(X ×B X,Q) belongs to the direct

summand H0(B, R4(π, π)∗Q) induced by the decomposition (4.3.43).

Admitting Proposition 4.3.13, the end of the proof of Theorem 4.3.10 is asfollows: By Theorem 4.1.3, we know that the relation

∆t = p∗1oXt · p∗23∆Xt + (perm.)− (p∗1oXt · p∗2oXt + (perm.))

holds in CH2(Xt,Q) for any t ∈ B. By Corollary 0.1.2, we conclude that thereexists a Zariski dense open set B0 of B such that (4.3.44) holds in H8(X ×B

X ×B X,Q). Proposition 4.3.13 thus implies Theorem 4.3.10.

Proof of Proposition 4.3.13. (i) We want to show that

πk ∪ (πi ⊗ πj) : Rπ∗Q⊗Rπ∗Q→ Rπ∗Q

vanishes for k 6= i + j.We note that

∪ : Rπ∗Q⊗Rπ∗Q→ Rπ∗Q

is induced, via the relative Kunneth decomposition

Rπ∗Q⊗Rπ∗Q ∼= R(π, π)∗Q

by the class [∆] of the small relative diagonal in X ×BX ×BX , seen as a relativecorrespondence between X ×B X and X , while P1 = π0, P2 = π4, P = π2 are


induced by the cycle classes [Z1], [Z2], [Z] ∈ H4(X ×BX ,Q), where Z := ∆X ⊂X ×B X . It thus suffices to show that the cycle classes

[Z2 ∆ (Z1 ×B Z1)], [Z ∆ (Z1 ×B Z1)],

[Z1 ∆ (Z ×B Z)], [Z ∆ (Z ×B Z)],

[Z1 ∆ (Z2 ×B Z2)], [Z ∆ (Z2 ×B Z2)], [Z2 ∆ (Z2 ×B Z2)],

[Z1 ∆ (Z1 ×B Z2)], [Z ∆ (Z1 ×B Z2)],

[Z1 ∆ (Z ×B Z2)], [Z ∆ (Z ×B Z2)], [Z2 ∆ (Z ×B Z2)],

[Z1 ∆ (Z1 ×B Z)], [Z2 ∆ (Z1 ×B Z)],

vanish in H8(X ×B X ×B X ,Q) over a dense Zariski open set of B. Here, allthe compositions of correspondences are over B.

Equivalently, it suffices to prove the following equality of cycle classes inH8(X 0 ×B X 0 ×B X 0,Q), X 0 = π−1(B0), for a Zariski dense open set of B0 ofB:

[∆] = [Z1 ∆ (Z1 ×B Z1)] + [Z2 ∆ (Z ×B Z)] + [Z ∆ (Z1 ×B Z)] (4.3.45)+[Z ∆ (Z ×B Z1)] + [Z2 ∆ (Z1 ×B Z2)] + [Z2 ∆ (Z2 ×B Z1)].

Replacing Z by ∆X − Z1 − Z2, we get

Z ×B Z = ∆X ×B ∆X −∆X ×B Z1 −∆X ×B Z2 − Z1 ×B ∆X

−Z2 ×B ∆X + Z1 ×B Z1 + Z2 ×B Z2 + Z1 ×B Z2 + Z2 ×B Z1

and thus (4.3.45) becomes

[∆] = [Z1 ∆ (Z1 ×B Z1)] + [Z2 ∆ (∆X ×B ∆X)] (4.3.46)−[Z2 ∆ (∆X ×B Z1)]− [Z2 ∆ (∆X ×B Z2)]− [Z2 ∆ (Z1 ×B ∆X )]−[Z2 ∆ (Z2 ×B ∆X )] + [Z2 ∆ (Z1 ×B Z1)] + [Z2 ∆ (Z2 ×B Z2)]+[Z2 ∆ (Z1 ×B Z2)] + [Z2 ∆ (Z2 ×B Z1)] + [Z ∆ (Z1 ×B ∆X )]−[Z ∆ (Z1 ×B Z1)]− [Z ∆ (Z1 ×B Z2)] + [Z ∆ (∆X ×B Z1)]−[Z ∆ (Z1 ×B Z1)]− [Z ∆ (Z2 ×B Z1)] + [Z2 ∆ (Z1 ×B Z2)]

+[Z2 ∆ (Z2 ×B Z1)].

We now have the following lemma:

Lemma 4.3.14. We have the following equalities of cycles in CH4(X ×BX ×B

X )Q (or relative correspondences between X ×B X and X )

∆ (Z1 ×B Z1) = p∗1oX · p∗2oX , (4.3.47)

∆ (∆X ×B ∆X ) = ∆, (4.3.48)


∆ (∆X ×B Z1) = p∗13∆X · p∗2oX , (4.3.49)

∆ (∆X ×B Z2) = p∗1oX · p∗3oX , (4.3.50)

∆ (Z1 ×B ∆X ) = p∗1oX · p∗23∆X , (4.3.51)

∆ (Z2 ×B ∆X ) = p∗2oX · p∗3oX , (4.3.52)

∆ (Z2 ×B Z2) = p∗3(oX · oX ), (4.3.53)

∆ (Z1 ×B Z2) = p∗1oX · p∗3oX , (4.3.54)

∆ (Z2 ×B Z1) = p∗2oX · p∗3oX , (4.3.55)

where the pi’s, for i = 1, 2, 3 are the projections from X ×B X ×B X to X andthe pij are the projections from X ×B X ×B X to X ×B X .

Proof. Equation (4.3.48) is obvious. Equations (4.3.47), (4.3.53), (4.3.54),(4.3.55) are all similar. Let us just prove (4.3.54). The cycle Z2 is X ×B oX ⊂X ×B X , and similarly Z1 = oX ×B X ⊂ X ×B X , hence Z1 ×B Z2 is the cycle

(oXb, x, y, oXb

), x ∈ Xb, y ∈ Xb, b ∈ B ⊂ X ×B X ×B X ×B X . (4.3.56)

(It turns out that in this case, we do not have to take care about the ordering wetake for the last inclusion.) Composing over B with ∆ ⊂ X ×B X ×B X is doneby taking the pull-back of (4.3.56) under p1234 : X 5/B → X 4/B , intersectingwith p∗345∆, and projecting the resulting cycle to X 3/B via p125. The resultingcycle is obviously

(oXb, x, oXb

), x ∈ Xb, b ∈ B ⊂ X ×B X ×B X ,

which proves (4.3.54).For the last formulas which are all of the same shape, let us just prove

(4.3.49). Recall that Z1 = oX ×B X ⊂ X ×B X . Thus ∆X ×B Z1 is the cycle

(x, x, oXb, y), x ∈ Xb, y ∈ Xb, b ∈ B ⊂ X ×B X ×B X ×B X .

But we have to see this cycle as a relative self-correspondence of X ×B X , forwhich the right ordering is

(x, oXb, x, y), x ∈ Xb, y ∈ Xb, b ∈ B ⊂ X ×B X ×B X ×B X . (4.3.57)

Composing over B with ∆ ⊂ X×BX×BX is done again by taking the pull-backof (4.3.57) by p1234 : X 5/B → X 4/B , intersecting with p∗345∆, and projectingthe resulting cycle to X 3/B via p125. Since ∆ = (z, z, z), z ∈ X , the consideredintersection is (x, oXb

, x, x, x), x ∈ Xb, b ∈ B, and thus the projection viap125 is (x, oXb

, x), x ∈ Xb, b ∈ B, thus proving (4.3.49).


Using Lemma 4.3.14 and the fact that the cycle p∗3(oX · oX ) vanishes bydimension reasons over a dense Zariski open set of B, (4.3.46) becomes, afterpassing to a Zariski open set of B if necessary:

[∆] = [Z1 (p∗1oX · p∗2oX )] + [Z2 ∆](4.3.58)−[Z2 (p∗13∆X · p∗2oX )]− [Z2 (p∗1oX · p∗3oX )]− [Z2 (p∗1oX · p∗23∆X )]

−[Z2 (p∗2oX · p∗3oX )] + [Z2 (p∗1oX · p∗2oX )] + [Z2 (p∗1oX · p∗3oX )]+[Z2 (p∗2oX · p∗3oX )] + [Z (p∗1oX · p∗23∆X )]− [Z (p∗1oX · p∗2oX )]−[Z (p∗1oX · p∗3oX )] + [Z (p∗13∆X · p∗2oX )]− [Z (p∗1oX · p∗2oX )]−[Z (p∗2oX · p∗3oX )] + [Z2 (p∗1oX · p∗3oX )] + [Z2 (p∗2oX · p∗3oX )],

which rewrites as

[∆] = [Z1 (p∗1oX · p∗2oX )] + [Z2 ∆] (4.3.59)−[Z2 (p∗13∆X · p∗2oX )]− [Z2 (p∗1oX · p∗23∆X )] + [Z2 (p∗1oX · p∗3oX )]

+[Z2 (p∗2oX · p∗3oX )] + [Z2 (p∗1oX · p∗2oX )] + [Z (p∗1oX · p∗23∆X )]−[Z (p∗1oX · p∗3oX )] + [Z (p∗13∆X · p∗2oX )]− 2[Z (p∗1oX · p∗2oX )].

To conclude, we use the following lemma:

Lemma 4.3.15. Up to passing to a dense Zariski open set of B, we have thefollowing equalities in CH4(X ×B X ×B X )Q:

Z1 (p∗1oX · p∗2oX ) = p∗1oX · p∗2oX , (4.3.60)

Z2 ∆ = p∗12∆X · p∗3oX , (4.3.61)

Z2 (p∗13∆X · p∗2oX ) = p∗2oX · p∗3oX , (4.3.62)

Z2 (p∗2oX · p∗3oX ) = p∗2oX · p∗3oX , (4.3.63)

Z2 (p∗1oX · p∗23∆X ) = p∗1oX · p∗3oX , (4.3.64)

Z2 (p∗1oX · p∗2oX ) = 0, (4.3.65)

Z2 (p∗1oX · p∗3oX ) = p∗1oX · p∗3oX , (4.3.66)

Z (p∗1oX · p∗3oX ) = 0, (4.3.67)

Z (p∗1oX · p∗23∆X ) = p∗1oX · p∗23∆X − p∗1oX · p∗2oX − p∗1oX · p∗3oX , (4.3.68)

Z (p∗13∆X · p∗2oX ) = p∗13∆X · p∗2oX − p∗1oX · p∗2oX − p∗2oX · p∗3oX , (4.3.69)

Z (p∗1oX · p∗2oX ) = 0. (4.3.70)


Proof. The proof of (4.3.62) is explicit, recalling that Z2 = (x, oXb), x ∈

Xb, b ∈ B, and that p∗13∆X · p∗2oX = (y, oXb, y), y ∈ Xb, b ∈ B. We then find

that Z2 (p∗13∆X · p∗2oX ) is the cycle

p124(p∗13∆X · p∗2oX · p∗34(Z2)) = p124((y, oXb, y, oXb

), y ∈ Xb, b ∈ B)= y, oXb

, oXb), y ∈ Xb, b ∈ B,

which proves (4.3.62). (4.3.63) is the same formula as (4.3.61) with the indices1 and 2 exchanged. The proofs of (4.3.60) to (4.3.66) work similarly.

For the other proofs, we recall that

Z = ∆X − Z1 − Z2 ⊂ X ×B X .

Thus we get, as ∆X acts as the identity:

Z (p∗1oX · p∗23∆X ) = p∗1oX · p∗23∆X −Z1 (p∗1oX · p∗23∆X )−Z2 (p∗1oX · p∗23∆X ).

We then compute the terms Z1 (p∗1oX ·p∗23∆X ) explicitly as before, which gives(4.3.68).

The other proofs are similar.

Using the cohomological version of Lemma 4.3.15, (4.3.59) becomes:

[∆] = [p∗1oX · p∗2oX ] + [p∗12∆X · p∗3oX ] (4.3.71)−[p∗2oX · p∗3oX )]− [p∗1oX · p∗3oX )] + [p∗1oX · p∗3oX ]

+[p∗2oX · p∗3oX ] + [p∗1oX · p∗23∆X − p∗1oX · p∗2oX − p∗1oX · p∗3oX ]+[p∗13∆X · p∗2oX − p∗1oX · p∗2oX − p∗2oX · p∗3oX ].

This last equality is now satisfied by assumption (compare with (4.3.44)) andthis concludes the proof of formula (4.3.45). Thus (i) is proved.

(ii) We just have to prove that

P1 ⊗ P1([∆X ]) = P2 ⊗ P2([∆X ]) = 0, (4.3.72)P1 ⊗ P ([∆X ]) = P2 ⊗ P ([∆X ]) = 0 in H4(X ×B X ,Q).

Indeed, the relative Kunneth decomposition gives

R(π, π)∗Q = Rπ∗Q⊗Rπ∗Q

and the decomposition (4.3.43) induces a decomposition of the above tensorproduct on the right:

Rπ∗Q⊗Rπ∗Q = ⊕k,lRkπ∗Q⊗Rlπ∗Q[−k − l], (4.3.73)

where the decomposition is induced by the various tensor products of P1, P2, P .Taking cohomology in (4.3.73) gives

H4(X ×B X ,Q) = ⊕s+k+l=4Hs(B, Rkπ∗Q⊗Rlπ∗Q).


The term H0(R4(π, π)∗Q) is then exactly the term in the above decompositionof H4(X ×B X ,Q) which is annihilated by the four projectors P1 ⊗P1, P1 ⊗P ,P2 ⊗ P , P2 ⊗ P2 and those obtained by changing the order of factors, whosevanishing will be deduced from the other ones by symmetry.

The proof of (4.3.72) is elementary. Indeed, consider for example the termP1 ⊗ P1, which is given by the cohomology class of the cycle

Z := pr∗1oX · pr∗2oX ⊂ X ×B X ×B X ×B X ,

which we see as a relative self-correspondence of X ×B X We have

Z∗(∆X ) = p34∗(p∗12∆X · Z).

But the cycle on the right is trivially rationally equivalent to 0 on fibers Xt×Xt.It thus follows from Corollary 0.1.2 that for some dense Zariski open set B0 ofB,

[Z]∗([∆X ]) = 0 in H4(X 0 ×B0 X 0,Q).

The other vanishing statements are proved similarly.

Chapter 5

Integral coefficients

Up to now, we have been working with rational coefficients, and indeed mostconjectures and statements made previously become wrong when passing toChow groups or Betti cohomology with integral coefficients. To start with, itis well-known since the work of Atiyah and Hirzebruch [4] that the Hodge con-jecture is not true for integral Hodge classes. What we would like to discuss inthis chapter is a number of birational invariants which can be defined using thisdefect. Of course, the important question is whether these birational invariantscan be nonzero for rationally connected on even unirational varieties.

5.1 Integral Hodge classes and birational invari-ants

5.1.1 Atiyah-Hirzebruch-Totaro topological obstruction.

Atiyah and Hirzebruch [4] found counterexamples to the Hodge conjecturestated for degree 2k integral Hodge classes (as opposed to rational Hodge classes),when k ≥ 2 . (We already mentioned that, in degree 2, the most optimisticstatements are true, due to the Lefschetz theorem on (1, 1)-classes.)

In [80], Totaro revisited these examples of Atiyah and Hirzebruch and re-formulated more directly the obstructions they had found, using the complexcobordism graded ring MU∗(X) of X. Let us first describe this ring which isdefined for all differentiable compact manifolds : given such an X, we considerfirst of all the objects which are triples (V, f, ε), where V is a differentiable com-pact manifold, f : V → X is a differentiable map, and ε is a class of a stablecomplex structure on the virtual normal bundle f∗TX − TV . Here f∗TX − TV

is an element of the K0-group of real vector bundles on V , and choosing a sta-ble complex structure on it means we choose an element of the K0-group ofcomplex vector bundles on V which sends (via the natural forgetful map) tof∗TX − TV + T , where T is the trivial real vector bundle of rank 1 or 0.)

One now makes the following construction, which is a slight variant of the

119

120 CHAPTER 5. INTEGRAL COEFFICIENTS

Thom construction of the absolute complex cobordism ring MU∗ = MU∗(point) :let us consider the free abelian group generated by such triples, and take itsquotient by the complex cobordism relations: namely, for each differentiablecompact manifold with boundary M , differentiable map φ : M → X, and stablecomplex structure ε on the virtual normal bundle f∗TX − TM , we observe thatthe restriction of TM to the boundary of M is naturally isomorphic to T∂M ⊕Twhere T is trivial of rank 1. Thus the stable complex structure ε on the virtualnormal bundle f∗TX − TM induces a stable complex structure ε0 on the virtualnormal bundle f∗0 TX − T∂M , where f0 is the restriction of f to ∂M .

We take the quotient of the free abelian group by the relations generated by

(∂M, f0, ε0) = 0,

(V1 t V2, f1 t f2, ε1 t ε2) = (V1, f1, ε1) + (V2, f2, ε2).

The result will be denoted by MU∗(X). Here the grading is given by ∗ =dimX − dimV . The product structure is given by fibered product over X.

Note that MU∗(X) is naturally a MU∗-module, since elements of MU∗ isgenerated by data (W, ε0) where W is differentiable compact and ε0 is a stablecomplex structure on TW (here the map to a point is necessarily constant).Then for (V, f, ε) and (W, ε0) as above, we can consider the product (V ×W, f pr1, ε + ε0).

Denote by MU∗(X) ⊗MU∗

Z its tensor product with Z over MU∗ (which maps

by the degree to Z = H0(point,Z)). Thus, in MU∗(X) ⊗MU∗

Z, one kills all the

products (V ×W, f pr1, ε1 + ε2) where ε2 is a stable complex structure on W ,with dimW > 0. As we have for such products

(f pr1)∗(1V×W ) = 0,

there is a natural map:

MU∗(X) ⊗MU∗

Z→ H∗(X,Z),

(V, f, ε) 7→ f∗1V .

(Here we note that, as we have a stable complex structure on the virtual normalbundle of f , the Gysin image f∗1V is well defined. If X is oriented, V isalso naturally oriented because the virtual normal bundle of f has a stablecomplex structure, and then f∗(1V ) is the Poincare dual cohomology class ofthe homology class f∗([V ]fund).)

Coming back to the case where X is a complex projective (or more generallycompact complex) manifold, Totaro [80] observed that the cycle map

Zk(X) → H2k(X,Z), Z 7→ [Z],

where the left hand side is the free abelian group generated by subvarieties (orirreducible closed analytic subsets) of codimension k of X, is the composite oftwo maps

Zk(X) → (MU∗(X) ⊗MU∗

Z)2k → H2k(X,Z). (5.1.1)

5.1. INTEGRAL HODGE CLASSES AND BIRATIONAL INVARIANTS 121

Here the second map was introduced above, and the first map is defined usingthe construction of the complex cobordism ring; indeed, for any inclusion ofa (maybe singular) codimension k closed algebraic subset Z ⊂ X, there is adesingularization

τ : Z → X,

and we have[Z] = τ∗1Z .

On the other hand, the virtual normal bundle of τ has an obvious stable complexstructure.

By the factorization (5.1.1), a torsion class in H2k(X,Z) which is not in theimage of

(MU∗(X) ⊗MU∗

Z)2k → H2k(X,Z)

cannot be algebraic. On the other hand, as it sends to 0 in H2k(X,C), it is ob-viously an integral Hodge class. This is a supplementary topological obstructionfor an integral Hodge class to be algebraic. These obstructions are of torsion,as the map (MU∗(X) ⊗

MU∗Z)2k → H2k(X,Z) becomes an isomorphism when

tensored by Q. In fact they essentially concern torsion classes, as explained byTotaro, as the map

(MU∗(X) ⊗MU∗

Z)2k → H2k(X,Z)

is an isomorphism if H∗(X,Z) has no torsion. The Atiyah-Hirzebruch example[4] shows that these obstructions are effective.

5.1.2 Kollar’s example

We start this section by describing a method due to Kollar [54], which producesexamples of smooth projective complex varieties X, together with an even degreeintegral cohomology class α, which is not algebraic, that is, which is not thecohomology class of an algebraic cycle of X, while a non-zero multiple of α isalgebraic. This is another sort of counterexample to the Hodge conjecture overthe integers, since the class α is of course a Hodge class.

The examples are as follows : consider a smooth hypersurface X ⊂ Pn+1 ofdegree d. For l < n, the Lefschetz theorem on hyperplane sections says that therestriction map

H l(Pn+1,Z) → H l(X,Z)

is an isomorphism. Since the left-hand side is isomorphic to ZHk for l = 2k < n,where H is the cohomology class of a hyperplane, and is 0 for l odd or l > n, weconclude by Poincare duality on X that for 2k > n, we have H2k(X,Z) = Zα,where α is determined by the condition < α, hn−k >= 1, with the notationh = H|X = c1(OX(1)). Note that the class d · α is equal to hk, (both haveintersection number d with hn−k), hence is algebraic.

In the sequel, we consider for simplicity the case where n = 3, k = 2. Thend · α is the class of a plane section of X.


Theorem 5.1.1. (Kollar, [54]) Assume that for some integer p coprime to 6,p3 divides d. Then for very general X, any curve C ⊂ X has degree divisible byp. Hence the class α is not algebraic.

Recall that “very general” means that the defining equation for X has tobe chosen away from a specified union of countably many Zariski closed propersubsets of the parameter space.

Proof. Let d = p3s, and let Y ⊂ P4 be a degree s smooth hypersurface. Letφ0, . . . , φ4 be sections of OP4(p) without common zeroes. They provide a map

φ : Y → P4,

which for a generic choice of the φi’s satisfies the following properties :

1. φ is generically of degree 1 onto its image, which is a hypersurface X0 ⊂ P4

of degree p3s = d.

2. φ is two-to-one generically over a surface in X0, three-to-one genericallyover a curve in X0, at most finitely many points of X0 have more than 3preimages, and no point has more than 4 preimages.

Let now X ⊂ P4 be a smooth hypersurface which is very general in mod-uli. Let C ⊂ X be a reduced curve. The idea is to degenerate the pair(X, C), C ⊂ X, to a pair (X0, C0), C0 ⊂ X0, where X0 was defined above.This is possible because the point parameterizing X is very general, and be-cause there are only countably many relative Hilbert schemes over the modulispace of X, parameterizing curves in the fibers Xt. Thus a curve C ⊂ X, with Xvery general in moduli, must correspond to a point of a relative Hilbert schemewhich dominates the moduli space of X.

By flatness, the curve C0 has the same degree as C. Recall the normalizationmap

φ : Y → X0.

By property 2 above, there exists a 1-cycle z0 in Y such that φ∗(z0) = 6z0,where z0 is the cycle associated to C0. It follows that

6deg z0 = deg φ∗(z0).

On the other hand, the right-hand side is equal to the degree of the line bundleφ∗OX0(1) computed on the cycle z0. Since φ∗OX0(1) is equal toOY (p), it followsthat this degree is divisible by p. Hence we found that 6deg C = 6deg C0 =6deg z0 is divisible by p, and since p is coprime to 6, it follows that deg C is alsodivisible by p.

As remarked in [78], Kollar’s example, which is not topological, exhibits thefollowing phenomenon which illustrates the complexity of the Hodge conjecture :we can have a family X → B of smooth projective complex manifolds, and a

5.2. RC VARIETIES AND THE RATIONALITY PROBLEM 123

locally constant integral Hodge class αt ∈ H4(Xt,Z), with the property thaton a dense (for the Euclidean topology) subset Balg ⊂ B, which is a countableunion of closed proper algebraic subsets of B, the class αt is algebraic, thatis, is an integral combination [Zt] =

∑i ni[Zi,t] of classes of codimension 2

subvarieties, but on its complementary set, which is a countable intersection ofdense open subsets, the class αt is not algebraic.

Indeed, one can show that there exists a countable union of proper algebraicsubsets, which is dense for the usual topology in the parameter space of allsmooth hypersurfaces of degree d in P4, consisting of points parameterizinghypersurfaces X for which the class α is algebraic. It suffices for this to provethat the set of smooth surfaces of degree d in P3 carrying an algebraic class λ ∈H2(S,Z) ∩ H1,1(S) satisfying the property that < λ, c1(OS(1)) >S is coprimeto d, is dense in the space of all surfaces of degree d in P3. Indeed, for any Xcontaining such a surface, the class α is algebraic on X.

Now this fact follows from the density criterion for the Noether-Lefschetzlocus explained in [86] II, 5.3.4, and from the fact that rational classes ν ∈H2(S,Q) such that a multiple bν is integral, and satisfies < bν, c1(OS(1)) >S= awith a coprime to d, are dense in H2(S,Q).

Remark 5.1.2. The argument above works only for the very general X, whichcould a priori exclude all varieties X defined over a number field K. HoweverHassett and Tschinkel gave an alternative argument (replacing the degenerationabove by specialization to an adequate closed fiber of a projective model ofX defined over SpecOK), showing that there exist varieties X satisfying theconclusions of Theorem 5.1.1 and defined over a number field.

5.2 Rationally connected varieties and the ra-tionality problem

A longstanding problem in algebraic geometry is the characterization of rationalvarieties, namely those smooth projective X which are birationally equivalentto Pn, n = dimX.

Beautiful obstructions to rationality, very efficient in dimension three, evenfor unirational varieties, (for which there exists a surjective rational map φ :Pn 99K X,) have been exhibited in the papers [18], [3], [48].

In higher dimensions, the criteria above, and especially those of [18], [3] areless useful. In [78], [89], we observed that if X is a smooth projective varietywhich is birational to Pn, then the Hodge conjecture holds for integral Hodgeclasses on X of degree 2n− 2 and 4. This is optimal, because in other degrees2i (different from 0, 2 and 2n where the groups are always 0), we can blow-upa copy of Kollar’s example imbedded in some projective space to get rationalvarieties X with nonalgebraic integral Hodge classes of degree 2i. In fact, wehave the following lemma (cf. [89]):


Lemma 5.2.1. The groups

Z4(X) := Hdg4(X,Z)/Hdg4(X,Z)alg, (5.2.2)Z2n−2(X) := Hdg2n−2(X,Z)/Hdg2n−2(X,Z)alg,

where the lower index “alg” means that we consider the group of integral Hodgeclasses which are algebraic, are both birational invariants of the complex projec-tive variety X of dimension n.

Proof. This follows from the resolution of singularities and the invarianceunder blow-ups, which is a consequence of the computation of the cohomologyand the Chow groups of a blown-up variety (cf [58], or [86], I, 7.3.3, II, 9.3.3).For the degree 4 case, the new degree 4 integral Hodge classes appearing underblow-up come from degree 2 (or degree 0) integral Hodge classes on the center ofthe blow-up. Hence they are algebraic by the Lefschetz theorem on (1, 1)-classes.For the other case, the new degree 2n−2 integral Hodge classes appearing underblow-up of a connected smooth subvariety are multiples of the class of a line ina fiber of the blowing-down map, hence they are also algebraic.

Note the following two facts concerning rational Hodge classes of degree 4and 2n− 2:

1. The Hodge conjecture is true for rational Hodge classes of degree 2n − 2on smooth projective varieties of dimension n. This is a consequence ofLefschetz theorem on (1, 1) classes, and of the hard Lefschetz theorem.

2. (cf. Section 2.1.2) The Hodge conjecture is true for rational Hodge classesof degree 4 on smooth projective varieties which have their CH0 groupsupported on a closed algebraic subset of dimension ≤ 3.

Thus it seems natural to consider in these situations the problem for integralHodge classes. The Kollar examples lead by taking products with Pr to otherexamples of non algebraic integral Hodge classes of degree 4 or 2n− 2, showingthat we have to restrict strongly the considered class of varieties.

Rationally connected varieties, for which there passes a rational curve throughany two points, have been the object of intensive work since the seminal paperby Kollar, Miyaoka and Mori [55]. Still they remain very mysterious from sev-eral points of view. They are as close as possible to unirational varieties (noexample is known to be not unirational), and this is a birationally and defor-mation invariant class. In view of the birational invariance explained above, itis thus natural to consider the problem of integral Hodge classes for them:

Question 5.2.2. Let X be a smooth rationally connected variety of dimensionn. Is the Hodge conjecture true for integral Hodge classes of degree 4 or 2n− 2on X?

It is tempting to believe that the answer should be “yes” for degree 2n− 2classes and we will give one theoretical argument in favour of this in Section5.2.1. We will show on the other hand in Section 5.2.2, following [22], that theanswer is negative in degree 4 for X of dimension ≥ 6.


5.2.1 The group Z2n−2(X)

For n = 3, we will have the equality 4 = 2n− 2, hence there is only one degreeto consider. In [88], we solved Question 5.2.2 in dimension 3, proving moregenerally the following result:

Theorem 5.2.3. Let X be a uniruled threefold, or a Calabi-Yau threefold,namely a threefold with trivial canonical bundle and vanishing irregularity. Thenthe Hodge conjecture is satisfied by integral Hodge classes of degree 4 on X, thatis, integral Hodge classes of degree 4 are generated by classes of curves in X.

Sketch of proof. The Hodge conjecture is satisfied by integral Hodgeclasses of degree 2. This is the Lefschetz theorem on (1, 1)-classes. Let α ∈H4(X,Z) and let j : Σ → X be the inclusion of a smooth ample surface intoX. The Lefschetz theorem on hyperplane restriction says that the Gysin map

j∗ : H2(Σ,Z) → H4(X,Z)

is surjective. Assume for simplicity that H2(X,OX) = 0 and that X is uniruled.Then there is no H3,1-part in the Hodge decomposition of H4(X,C) and thuswe want to show that any integral degree 4 cohomology class is algebraic, thatis, is a combination with integral coefficients of classes of curves in X. The keypoint (in the uniruled case) is then the following:

Proposition 5.2.4. If Σ is chosen ample enough (that is belongs to the linearsystem associated to a sufficiently high power of an ample line bundle on X)and satisfies the condition that

Σ2.c1(KX) < 0,

then H2(Σ,Z) is generated over Z by classes which become algebraic on Σt,where Σt is a small deformation of Σ in X.

Note that if X is uniruled, after performing a birational transformation ofX, we can construct a smooth projective variety X ′ with an ample line bundleL such that c1(L)2 · c1(KX′) < 0. Thus, up to replacing X by X ′, we mayassume surfaces Σ as above exist.

If now α ∈ H2(Σ,Z) becomes algebraic under a small deformation of Σ inX, this means that for a deformation jt : Σt → X, the class αt deduced from αby flat transport satisfies αt = [Zt] ∈ H2(Σt,Z), for some divisor Zt ∈ CH1(Σt)and thus

j∗α = jt∗αt = jt∗[Zt] = [jt∗(Zt)] ∈ H4(X,Z).

Thus it follows from the surjectivity of j∗ and from Proposition 5.2.4 thatH4(X,Z) is generated by classes of 1-cycles on X.

The proof of Proposition 5.2.4 uses the study of infinitesimal variations ofHodge structures. Using the Lefschetz theorem on (1, 1)-classes, one has equiv-alently to prove that H2(Σ,Z) is generated over Z by classes which become of


type (1, 1) under a small deformation of Σ in X. Thus, this is mainly a questionof showing that the spaces

H1,1(Σ)R := H1,1(Σ) ∩H2(Σ,R)

“move enough” with Σ ⊂ X inside H2(Σ,R), so as to fill-in an open subsetV = ∪t∈UH1,1(Σt) ∩ H2(Σt,R) ⊂ H2(Σ,R). Here U is a simply connectedopen set of the space of smooth deformations of Σ in X, and we freely use thecanonical identification H2(Σt,R) ∼= H2(Σ,R) given by parallel transport.

As this open subset V is a cone, it will then be clear that integral points inthis cone will generate over Z the whole lattice H2(Σ,Z).

The study of the deformations of the subspace H1,1(Σ)R ⊂ H2(Σ,R) isdone using Griffiths machinery of infinitesimal variations of Hodge structuresfor hypersurfaces (cf. [40], [86], II, 6.2).

Remark 5.2.5. It would be tempting to weaken the assumptions in Theorem5.2.3 and to ask whether a smooth projective 3-fold X with trivial CH0 group(or CH0 group supported on a surface) satisfies the condition Z4(X) = 0.This is essentially disproved in [22], (except that we do not know that theexample we have indeed satisfies the conclusion that CH0(X) is trivial). Infact, we produce following Kollar an example of a smooth projective threefoldX satisfying Hi(X,OX) = 0, i > 0 and with non trivial Z4(X). By the Blochconjecture 2.2.4 (and Roitman’s theorem [73]), this X should satisfy CH0(X) =Z.

In dimension ≥ 4, Question 5.2.2 has been studied in [44]. We prove thefollowing result:

Theorem 5.2.6. (Horing-Voisin 2010) Let X be a Fano fourfold or a Fanofivefold of index 2. Then X satisfies Z2n−2(X) = 0.

For the proof, we first extend the Calabi-Yau case in Proposition 5.2.3 al-lowing X to have isolated canonical singularities. We then prove that a Fanofourfold contains an anticanonical divisor with isolated canonical singularities.The case of a Fano fivefold of index 2 works similarly: If −KX = 2H, whereH ∈ Pic X is an ample line bundle, we show that there is a complete intersectionof two members of |H| which is a threefold with isolated canonical singularities,and of course with trivial canonical bundle.

To conclude this section, let us prove the following result:

Theorem 5.2.7. (cf. [97]) The group Z2n−2(X) is locally deformation invari-ant for rationally connected n-folds. In particular its order is a deformationinvariant.

Let us first explain the meaning of the statement. Consider a smooth projec-tive morphism π : X → B between connected quasi-projective complex varieties,with n dimensional fibers. Recall from [55] that if one fiber Xb := π−1(b) is ra-tionally connected, so is every fiber. Let us endow everything with the usual


topology. Then the sheaf R2n−2π∗Z is locally constant on B. On any Euclideanopen set U ⊂ B where it is trivial, the group Z2n−2(Xb), b ∈ U is the finitequotient of the constant group H2n−2(Xb,Z) by its subgroup H2n−2(Xb,Z)alg.To say that Z2n−2(Xb) is locally constant means that on open sets U as above,the subgroup H2n−2(Xb,Z)alg of the constant group H2n−2(Xb,Z) does notdepend on b.

Proof of Theorem 5.2.7. We first observe that, due to the fact thatrelative Hilbert schemes parameterizing curves in the fibers of B are a countableunion of varieties which are projective over B, given a simply connected openset U ⊂ B (in the classical topology of B), and a class α ∈ Γ(U,R2n−2π∗Z) suchthat αt is algebraic for t ∈ V , where V is a smaller nonempty open set V ⊂ U ,then αt is algebraic for any t ∈ U .

To prove the deformation invariance, we just need using the above observa-tion to prove the following:

Lemma 5.2.8. Let t ∈ U ⊂ B, and let C ⊂ Xt be a curve and let [C] ∈H2n−2(Xt,Z) be its cohomology class. Then the class [C]s is algebraic for s inU .

Proof of Lemma 5.2.8. By results of [55], there are rational curves Ri ⊂Xt with ample normal bundle which meet C transversally at distinct points, andwith arbitrary tangent directions at these points. We can choose an arbitrarilylarge number D of such curves with generically chosen tangent directions at theattachment points. We then know by [55] that the curve C ′ = C ∪i≤D Ri issmoothable and that the result is a smooth unobstructed curve C ′′ ⊂ Xt, thatis H1(C ′′, NC′/Xt

) = 0. This curve C ′′ then deforms with Xt (cf. [52], [53, II.1])in the sense that the morphism from the deformation of the pair (C ′′, Xt) to Bis smooth. So for s ∈ U , there is a curve C ′′s ⊂ Xs which is a deformation ofC ′′ ⊂ Xt. The class [C ′′s ] = [C ′′]s is thus algebraic on Xs. On the other hand,we have

[C ′′] = [C ′] = [C] +∑

i

[Ri].

As the Ri’s are rational curves with positive normal bundle, they are also un-obstructed, so that the classes [Ri]s also are algebraic on Xs. Thus [C]s =[C ′′]s −

∑i[Ri]s is algebraic on Xs. The lemma, hence also the theorem, is

proved.

Remark 5.2.9. The same proof applies to prove invariance of the group Z2n−2(X)for X a rationally connected variety defined over a number field K, under spe-cialization (in an adequate sense) to a point p of SpecOK such that Xp issmooth, where X is a projective model of X over SpecOK . We refer to [97] forthis result and the following Theorem 5.2.10, (which strongly suggests in factthe vanishing of Z2n−2(X) for X a rationally connected variety over C).


Theorem 5.2.10. Assume the Tate conjecture is true for divisor classes onvarieties over a finite field (cf. [59]). Then the group Z2n−2(X) vanishes forany rationally connected n-fold over C.

5.2.2 The group Z4(X) and unramified cohomology

Let X be a smooth projective complex variety. We will denote Xcl the setX(C) endowed with its classical (or Euclidean) topology, and XZar the setX(C) endowed with its Zariski topology.

Letπ : Xcl → XZar

be the identity of X(C). This is obviously a continuous map, and Bloch-Ogustheory [12] is the study of the Leray spectral sequence associated to this mapand any constant sheaf with stalk A on Xcl. The abelian group A will be inapplications one of the following groups: Z, Q, Q/Z.

We are thus led to introduce the sheaves on XZar

Hi(A) := Riπ∗A.

By definition, Hi(A) is thus the sheaf associated to the presheaf U 7→ HiB(U,A)

on XZar.The Leray spectral sequence for π and A has terms

Ep,q2 = Hp(XZar,Hq(A)).

Unramified cohomology of X with value in A is defined by the formula (cf.[20])

Hinr(X,A) = H0(XZar,Hi

X(A)).

The main result of the paper by Bloch and Ogus [12] is the following Gersten-Quillen resolution for the sheaves Hi

X(A). For any closed subvariety D ⊂ X,let iD : D → X be the inclusion map and Hi(C(D), A) the constant sheaf on Dwith stalk

lim→

U⊂D

nonempty Zariski open

Hi(U(C), A)

at any point of D. When D′ ⊂ D has codimension 1, there is a map inducedby the topological residue (on the normalization of D) (cf. [86, II, 6.1.1]) :

ResD,D′ : Hi(C(D), A) → Hi−1(C(D′), A).

For r ≥ 0, let X(r) be the set of irreducible closed algebraic subsets ofcodimension r in X.

Theorem 5.2.11. ([12], Theorem 4.2) For any A, and any integer i ≥ 1, thereis an exact sequence of sheaves on XZar

0 → HiX(A) → iX∗Hi(C(X), A) ∂→

⊕

D∈X(1)

iD∗Hi−1(C(D), A) ∂→


. . .∂→

⊕

D∈X(i)

iD∗AD → 0.

Here the components of the maps ∂ are induced by the maps ResD,D′ whenD′ ⊂ D (and are 0 otherwise). The sheaf AD on DZar identifies of course tothe constant sheaf with stalk H0(C(D), A).

Let us state a few consequences proved in [12]: First of all, denoting byCHk(X)/alg the group of codimension k cycles of X modulo algebraic equiva-lence, we get the Bloch–Ogus formula:

Corollary 5.2.12. ([12], Corollary 7.4) If X is a smooth complex projectivevariety, there is a canonical isomorphism

CHk(X)/alg = Hk(XZar,Hk(Z)). (5.2.3)

Proof. Indeed, the Bloch–Ogus resolution is acyclic. It thus allows tocompute Hk(XZar,Hk(Z)) by taking global sections in the above resolution,which gives

Hk(XZar,Hk(Z)) = Coker (∂ : ⊕D∈X(k−1)H1(C(W ),Z) −→ ⊕D∈X(k)Z)

The group⊕D∈X(k)Z is the group of codimension k cycles on X, and to conclude,one has to check that the image of the map ∂ above is the group of cyclesalgebraically equivalent to 0. This follows from the fact that on a smoothprojective variety W , a divisor D is cohomologous to zero (hence algebraicallyequivalent to zero) if and only if there exists a degree 1 cohomology class α ∈H1(W \ SuppD,Z) such that Resα = D.

By theorem 5.2.11, the sheaf Hi(A) has an acyclic resolution of length ≤ i.We thus get the following vanishing result:

Corollary 5.2.13. For X smooth, A an abelian group and r > i, one has

Hr(XZar,HiX(A)) = 0. (5.2.4)

Concerning the structure of the sheaves Hi(Z), we have the following result,which is a consequence of the Bloch-Kato conjecture recently proved by Rost andVoevodsky (we refer to [14], [22], [5] for more explanations concerning the waythe very important result below is deduced from the Bloch-Kato conjecture).

Theorem 5.2.14. The sheaves Hi(Z) of Z-modules on XZar have no torsion.

The following Corollary gives an equivalent formulation of this theorem, byconsidering the long exact sequence associated to the short exact sequence ofsheaves on Xcl

0 → Z→Q→ Q/Z→ 0

on Xcl and the associated long exact sequence of sheaves on XZar

. . . → Hi(Q) → Hi(Q/Z) → Hi+1(Z)→Hi+1(Q) → . . . .


Corollary 5.2.15. For any integer i, there is a short exact sequence of sheaveson ZZar

0 → Hi(Z) → Hi(Q) → Hi(Q/Z) → 0.

Unramified cohomology with torsion coefficients (that is, A = Z/nZ orA = Q/Z) plays an important role in the study of the Luroth problem, thatis the study of unirational varieties which are not rational, (see for examplethe papers [61], [20], [71]). In fact, the invariant used by Artin and Mumford,which is the torsion in the group H3

B(X,Z), is equal for rationally connectedvarieties to the unramified cohomology group H2

nr(X,Q/Z). In the paper [20],the authors exhibit unirational sixfolds with vanishing group H2

nr(X,Q/Z) butnon vanishing group H3

nr(X,Q/Z). Their example is reinterpreted in the recentpaper [22], using the groups Z4(X) introduced in (5.2.2).

More precisely, we give in the paper [22] the following comparison resultbetween Z4(X) and H3

nr(X,Q/Z) (a similar result was in fact also establishedin [5]):

Theorem 5.2.16. (Colliot-Thelene and Voisin 2010) (1) For any smooth pro-jective X, there is an exact sequence

0 → H3nr(X,Z)⊗Q/Z→ H3

nr(X,Q/Z) → Tors(Z4(X)) → 0.

(2) If CH0(X) is supported on a closed algebraic subset of dimension ≤ 3,Tors(Z4(X)) = Z4(X). If CH0(X) is supported on a closed algebraic subset ofdimension ≤ 2, H3

nr(X,Z) = 0.(3) In particular, if X is rationally connected (so that CH0(X) = Z), we

haveH3

nr(X,Q/Z) ∼= Z4(X).

As a consequence, we get a negative answer to Question 5.2.2 for degree4 integral Hodge classes on certain rationally connected varieties (and evenunirational varieties) of dimension at least 6.

Theorem 5.2.17. The Colliot-Thelene-Ojanguren varieties X constructed in[20], which are unirational 6-folds, hence rationally connected, have Z4(X) 6= 0.

Indeed, as proved in [20], these varieties have a non trivial unramified coho-mology group H3

nr(X,Q/Z).

Sketch of proof of Theorem 5.2.16. Assuming that the Hodge conjectureis satisfied for degree 2i rational Hodge classes on X, the group Z2i(X) is oftorsion. The fact that the Hodge conjecture is satisfied for degree 4 rationalHodge classes on X if X has the property that CH0(X) is supported on aclosed algebraic subset of dimension ≤ 3 is Theorem 2.1.11 (cf. Bloch-Srinivas[14]). This proves the first statement of (2). The second statement in (2) isproved by applying the Bloch-Srinivas decomposition of the diagonal (Theorem2.1.7) and letting act both sides on unramified cohomology (cf. [22, Appendix]).


It thus follows that H3nr(X,Z) is of torsion. As it has no torsion by Theorem

5.2.14, it is in fact 0.We now prove (1). We observe first of all that the torsion of the group

Z2i(X) = Hdg2i(X,Z)/H2i(X,Z)alg is always isomorphic to the torsion of thegroup H2i(X,Z)/H2i(X,Z)alg. We examine now the Bloch-Ogus spectral se-quence of X with coefficients in Z. The E2-terms in degree 4 are

H2(XZar,H2(Z)) = E2,22 , H1(XZar,H3(Z)) = E1,3

2 , H0(XZar, H4(Z)) = E0,42 .

No non zero differential dr, r ≥ 2 starts from H2(XZar,H2(Z)) or H1(XZar,H3(Z))due to Corollary 5.2.13. No nonzero differential dr, r ≥ 2 arrives at H1(XZar,H3(Z))or H0(XZar, H4(Z)) for degree reasons. Thus we have

E2,22 ³ E2,2

∞ , E1,32 = E1,3

∞ , E0,42 ⊂ E0,4

∞ .

It follows that there is a natural composed map

H2(XZar,H2(Z)) → E2,2∞ ⊂ H4(X,Z),

whose image is the deepest level N2H4(X,Z) of the Leray-Bloch-Ogus filtra-tion (which is in fact the coniveau filtration). This map identifies to the cycleclass map (cf. [12]), so that its cokernel H4(X,Z)/N2H4(X,Z) is the groupH4(X,Z)/H4(X,Z)alg. This group has a filtration induced by the Leray spec-tral sequence, and the graded pieces are

E1,3∞ = H1(XZar,H3(Z)), E0,4

∞ ⊂ H0(XZar, H4(Z)).

By Theorem 5.2.14, the group H0(XZar, H4(Z)) has no torsion, hence it followsfinally that we have an isomorphism:

Tors (H4(X,Z)/H4(X,Z)alg) = Tors (H1(XZar,H3(Z))).

The group on the left identifies to the group Tors (Z4(X)) by the observationabove, while the group on the right is analyzed by the exact sequence of sheaveson XZar given in Corollary 5.2.15:

0 → H3(Z) → H3(Q) → H3(Q/Z) → 0..

The induced long exact sequence immediately gives

Tors (H1(XZar,H3(Z))) ∼= H0(X,H3(Q/Z))/Im (H0(X,H3(Q))),

which finishes the proof.

Remark 5.2.18. The same proof shows that the torsion subgroup TorsH3(X,Z)identifies to the unramified cohomology group H2

nr(X,Q/Z) for a variety Xwhose CH0 group is supported on a curve.


In the paper [96], we give a similar cycle-theoretic interpretation of thegroup H4

nr(X,Q/Z) (Theorem 5.2.21 below). We need for this to introduce afew notations:

Let X be a smooth complex projective variety and k, l be integers. We havethe subgroup N lHk

B(X,Z) ⊂ HkB(X,Z) of “coniveau l cohomology”, defined as

N lHkB(X,Z) = Ker (Hk

B(X,Z) → lim→

codim W=l

HkB(X \W,Z)),

where the W ⊂ X considered here are the closed algebraic subsets of X ofcodimension l. Introducing a resolution of singularities W of W and its naturalmorphism τW : W → X to X, we have

N lHkB(X,Q) =

∑

codim W=l

Im (τW∗ : Hk−2lB (W ,Q) → Hk(X,Q)),

as explained in the proof of Theorem 1.2.20.From now on, we restrict to the case k = 2l + 1. For any W ⊂ X, τW :

W → X as above, the Gysin morphism τW∗ : H1B(W ,Z) → H2l+1

B (X,Z) is amorphism of Hodge structures (of bidegree (l, l)), which induces a morphismbetween the intermediate Jacobians (cf. [86, I,12.2])

τW∗ : Pic0(W ) = J1(W ) → J2l+1(X) :=H2l+1

B (X,C)F lH2l+1

B (X,C)⊕H2l+1B (X,Z)/torsion

.

This map τW∗ is compatible in an obvious way with the Abel-Jacobi maps φW

and φX , defined respectively on codimension 1 and codimension l + 1 cycles ofW and X which are homologous to 0.

The Deligne cycle class map

cll+1D : CH l+1(X) → H2l+2

D (X,Z(l + 1))

restricts to the Abel-Jacobi map φlX on the subgroup of cycles homologous to

0 (cf. [86, I,12.3.3]), and in particular on the subgroup of cycles algebraicallyequivalent to 0.

If Z ∈ CH l(X) is algebraically equivalent to 0, there exist subvarietiesWi ⊂ X of codimension l and cycles Zi ⊂ Wi homologous to 0 such thatZ =

∑i τWi∗Zi in CH l+1(X). It follows from the previous considerations that

cll+1D induces a morphism

cll+1D,tr : CH l(X)/alg → H2l+2

D (X,Z(l + 1))tr :=H2l+2D (X,Z(l + 1))

〈τW∗J1(W ), codim W = l〉.

Let T l+1(X) := Tors (Ker cll+1D,tr).

Lemma 5.2.19. This group identifies to the image of the subgroup Tors (Ker cll+1D )

in CH l+1(X)/alg.

5.3. INTEGRAL DECOMPOSITION OF THE DIAGONAL 133

Proof. This follows from the fact that the groups of cycles algebraicallyequivalent to 0 modulo rational equivalence are divisible. This implies that thenatural map Tors(CHi(X)) → Tors(CHi(X)/alg) is surjective for any i. Wethen use the fact that for the W ’s introduced above, the map

CH1(W )alg∼= Pic0(W ) → J1(W )

is an isomorphism, where CH1(W )alg ⊂ CH1(W ) is the subgroup of cyclesalgebraically equivalent to 0.

We have the following lemma :

Lemma 5.2.20. The group T 3(X) is a birational invariant of X.

Proof. It suffices to check invariance under blow-up. The Manin formulas(cf. [58], [86, II,9.3.3]) for groups of cycles modulo rational or algebraic equiv-alence and for Deligne cohomology of a blow-up imply that it suffices to provethat the groups T i(Y ) are trivial for i ≤ 2 and Y smooth projective. Howeverthis is an immediate consequence of the definition, of Lemma 5.2.19, and of thefact that the Deligne cycle class map clD : CHi(X) → H2i

D (X,Z(i)) is injectiveon torsion cycles of codimension i ≤ 2 (cf. [63]).

The following interpretation of degree 4 unramified cohomology with finitecoefficients is proved in [96]:

Theorem 5.2.21. (Voisin 2011) Assume that the group H5B(X,Z)/N2H5

B(X,Z)has no torsion. Then the quotient of H4

nr(X,Q/Z) by H4nr(X,Z)⊗Q/Z identifies

to the group T 3(X).Equivalently, there is an exact sequence

0 → H4nr(X,Z)⊗Q/Z→ H4

nr(X,Q/Z) → Tors (CH3(X)/alg)cl3D,tr→ H6

D(X,Z(3))tr.

One can of course add that for a variety X with CH0(X) supported on a 3-dimensional closed algebraic subset X ′ ⊂ X, (for example a rationally connectedvariety), the group H4

nr(X,Z)⊗Q/Z on the left is 0.

5.3 Integral decomposition of the diagonal andthe structure of the Abel-Jacobi map

5.3.1 Structure of the Abel-Jacobi map and birational in-variants

Recall the diagonal decomposition principle (Theorem 2.1.7) which says that ifY is a smooth variety such that CH0(Y ) = Z, there is an equality in CHd(Y ×Y ), d = dimY :

N∆Y = Z1 + Z2, (5.3.5)



SuppZ1 ⊂ D × Y, D Y, Z2 = N(Y × pt).

Note that the integer N appearing above cannot in general be set equal to1, and one purpose of this concluding section is to investigate the significanceof this invariant, at least if we work on the level of cycles modulo homologicalequivalence. We will say that Y admits a cohomological decomposition of thediagonal as in (5.3.5) if there is an equality of cycle classes

N [∆Y ] = [Z1] + [Z2],

in H2d(Y × Y,Z), with Supp Z1 ⊂ D × Y and Z2 = N(Y × pt). We will saythat Y admits an integral cohomological decomposition of the diagonal if thereis such a decomposition with N = 1.

Remark 5.3.1. The minimal positive integers N such that a decompositionas above exists in either the Chow-theoretic or the cohomological setting arebirational invariants of Y . Indeed, if we have a birational morphism

φ : X → Y,

which is an isomorphism onto its image away from a divisor E ⊂ X, then wehave in CHd(X ×X)

∆X = φ∗(∆Y ) + Z,

where Z is a cycle supported on E × E. As the term Z is absorbed in the termZ1 of a diagonal decomposition for X, the same integers N work for X and Y .

Recall that the existence of a cohomological decomposition of the diagonalas above has strong consequences (see [14] and sections 2.1.1 and 2.1.2):

1. This implies the generalized Mumford theorem which says in this case thatHi(Y,OY ) = 0 for i ≥ 1. In particular the Hodge structures on H2(Y,Q),hence on its Poincare dual H2d−2(Y,Q) are trivial.

2. The group Z4(Y ) := Hdg4(Y )/Hdg4(Y )alg is of torsion (annihilated bythe integer N above).

3. The intermediate Jacobian J3(Y ) defined by

J(Y ) := H3(Y,C)/(F 2H3(Y )⊕H3(Y,Z)). (5.3.6)

is thus an abelian variety (cf. [40], [86, I,12.2.2]) and the Abel-Jacobi mapCH2(Y )hom → J3(Y ) is surjective with finite kernel (annihilated by theinteger N above).

Concerning the last point, much more is true assuming the existence of a Chowtheoretic decomposition of the diagonal.


Theorem 5.3.2. (cf. [14], [63]) If Y is a smooth complex projective varietysuch that CH0(Y ) = Z, then the Abel-Jacobi map induces an isomorphism :

AJY : CH2(Y )hom = CH2(Y )alg∼= J(Y ). (5.3.7)

Remark 5.3.3. In fact the concusion if we only assume that CH0(Y ) is sup-ported on a curve.

This theorem is proved by delicate arguments from algebraic K-theory in-volving the Merkurjev-Suslin theorem (the degree 2 case of the Bloch-Kato con-jecture), Gersten-Quillen resolution in K-theory, and Bloch-Ogus theory [12].The statement that under the above assumptions CH2(Y )hom = CH2(Y )alg

is obtained in [14], while it is true without any assumptions on Y that theAbel-Jacobi map is injective on torsion codimension 2 cycles in CH2(Y ).

The group on the left in (5.3.7) a priori does not have the structure of analgebraic variety, unlike the group on the right which is an abelian variety.However it makes sense to say that AJY is algebraic, with the meaning that forany smooth algebraic variety B, and any codimension 2-cycle Z ⊂ B × Y , withZb ∈ CH2(Y )hom for any b ∈ B, the induced map

φZ : B → J(Y ), b 7→ AJY (Zb),

is a morphism of algebraic varieties.Consider the case of a uniruled 3-fold Y with CH0(Y ) = Z. Then we also

have Theorem 5.2.3 saying that the integral degree 4 cohomology H4(Y,Z) isgenerated over Z by classes of curves, and thus the birational invariant

Z4(Y ) :=Hdg4(Y,Z)

< [Z], Z ⊂ Y, codim Z = 2 >

introduced in (5.2.2) is trivial in this case.The conclusion of the above mentioned theorems 5.3.2 and 5.2.3 is that for a

uniruled threefold with CH0 = Z, all the interesting (and birationally invariant)phenomena concerning codimension 2 cycles, namely the kernel of the Abel-Jacobi map (Mumford [61]), the Griffiths group (Griffiths [40]) and the groupZ4(X) versus degree 3 unramified cohomology with torsion coefficients (Soule-Voisin [78], Colliot-Thelene-Voisin [22]) are trivial. In the rationally connectedcase, the only interesting cohomological or Chow theoretic invariant could bethe Artin-Mumford invariant (or degree 2 unramified cohomology with torsioncoefficients, cf. [20]), which is also equal to the Brauer group since H2(Y,OY ) =0.

Still the geometric structure of the Abel-Jacobi map on families of 1-cycleson such threefolds is mysterious, in contrast to what happens in the curve case,where Abel’s theorem shows that the Abel-Jacobi map on the family of effective0-cycles of large degree has fibers isomorphic to projective spaces.

There are for example two natural questions (Questions 5.3.4 and 5.3.7) leftopen by Theorem 5.3.2 :


Question 5.3.4. Let Y be a smooth projective threefold, such that AJY :CH1(Y )alg → J(Y ) is surjective. Is there a codimension 2 cycle Z ⊂ J(Y )× Ywith Zb ∈ CH2(Y )hom for b ∈ J(Y ), such that the induced morphism

φZ : J(Y ) → J(Y ), φZ(b) := AJY (Zb)

is the identity?

Note that the surjectivity assumption is conjecturally implied by the van-ishing H3(Y,OY ) = 0, via the generalized Hodge conjecture 1.2.21 or by theHodge conjecture for degree 4 rational Hodge classes on Y × J(Y ) (cf. proof ofTheorem 1.2.23).

Remark 5.3.5. One can more precisely introduce a birational invariant of Ydefined as the gcd of the non zero integers N for which there exist a variety Band a cycle Z ⊂ B × Y as above, with deg φZ = N . Question 5.3.4 can then bereformulated by asking whether this invariant is equal to 1.

Remark 5.3.6. Question 5.3.4 has a positive answer if the Hodge conjecturefor degree 4 integral Hodge classes on Y × J(Y ) are algebraic. Indeed, theisomorphism H1(J(C),Z) ∼= H3(Y,Z) is an isomorphism of Hodge structureswhich provides a degree 4 integral Hodge class α on J(Y )× Y (see the proof ofTheorem 1.2.23 or [86, I,Lemma 11.41]). A codimension 2 algebraic cycle Z onJ(Y )× Y with [Z] = α would provide a solution to Question 5.3.4.

The following question is an important variant of the previous one, whichappears to be much more natural from a geometric point of view.

Question 5.3.7. Is the following property (*) satisfied by Y ?(*) There exist a smooth projective variety B and a codimension 2 cycle

Z ⊂ B × Y , with Zb ∈ CH2(Y )hom for any b ∈ B, such that the inducedmorphism φZ : B → J(Y ) is surjective with rationally connected general fiber.

This question has been solved by Iliev-Markushevich and Markushevich-Tikhomirov ([45], [60], see also [43] for similar results obtained independently)in the case where Y is a smooth cubic threefold in P4. The answer is alsoaffirmative for the intersection X of two quadrics in P5 (cf. [72]): in this casethe family of lines in X is a surface isomorphic via a choice of base point to theintermediate Jacobian J(X).

Obviously a positive answer to Question 5.3.4 implies a positive answer toQuestion 5.3.7, as we can then just take B = J(X). We will provide a moreprecise relation between these two questions in section 5.3.2. However, it seemsthat Question 5.3.7 is more natural, especially if we go to the following strongerversion (Question 5.3.8):

Here we choose an integral cohomology class α ∈ H4(Y,Z). AssumingCH0(Y ) is supported on a curve, the Hodge structure on H4(Y,Q) is trivialand thus α is a Hodge class. Introduce the torsor J(Y )α defined as follows: theDeligne cohomology group H4

D(Y,Z(2)) is an extension

0 → J(Y ) → H4D(Y,Z(2)) o→ Hdg4(Y,Z) → 0, (5.3.8)


where o is the natural map from Deligne to Betti cohomology (cf. [86, I,Corollary12.27]). Define

J(Y )α := o−1(α). (5.3.9)

By definition, the Deligne cycle class map (cf. [33], [86, I,12.3.3]), restricted tocodimension 2 cycles of class α, takes value in J(Y )α. Furthermore, for anyfamily of 1-cycles Z ⊂ B × Y of class [Zb] = α, b ∈ B, parameterized by analgebraic variety B, the map φZ induced by the Abel-Jacobi map (or rather theDeligne cycle class map) of Y , that is

φZ : B → J(Y )α, φZ(b) = AJY (Zb),

is a morphism of complex algebraic varieties (note again that the twisted com-plex torus J(Y )α is algebraic because H3,0(Y ) = 0, cf. [86, I,12.2.2]). Thefollowing question makes sense for any smooth projective threefold Y satisfyingthe conditions H2(Y,OY ) = H3(Y,OY ) = 0:

Question 5.3.8. Is the following property (**) satisfied by Y ?(**) For any degree 4 integral cohomology class α on Y , there is a “natu-

rally defined” (up to birational transformations) smooth projective variety Bα,together with a codimension 2 cycle Zα ⊂ Bα × Y , with [Zα,b] = α in H4(Y,Z)for any b ∈ B, such that the morphism φZα : Bα → J(Y )α is surjective withrationally connected general fiber.

By “naturally defined”, we have in mind that Bα should be determined byα by some natural geometric construction (eg., if α is sufficiently positive, adistinguished component of the Hilbert scheme of curves of class α and givengenus, or a moduli space of vector bundles with c2 = α), which would implythat Bα is defined over the same definition field as Y .

This question is solved affirmatively by Castravet in [16] when Y is thecomplete intersection of two quadrics. It is also solved affirmatively in [93]for cubic threefolds. Let us comment on the importance of Question 5.3.8 inrelation with the Hodge conjecture with integral coefficients for degree 4 Hodgeclasses, (that is the study of the group Z4 introduced in (5.2.2)): The importantpoint here is that we want to consider fourfolds fibered over curves, or familiesof threefolds Yt parameterized by a curve Γ. The generic fiber of this fibrationis a threefold Y defined over C(Γ). Property (**) essentially says that property(*), being satisfied over the definition field, which is in this case C(Γ), holds infamily. When we work in families, the necessity to look at all torsors J(Y )α,and not only at J(Y ), becomes obvious: for fixed Y the twisted Jacobians areall isomorphic (maybe not canonically) and if we can choose a cycle zα in eachgiven class α (for example if Y is uniruled so that Z4(Y ) = 0), we can usetranslations by the zα to reduce the problem to the case where α = 0; thisis a priori not true in families, for example because there might not be anycodimension 2 cycle of relative class α on the total space of the family.

We have the following application (Theorem 5.3.9) of this property (**) tothe Hodge conjecture for degree 4 integral Hodge classes on fourfolds fibered


over curves. Let X be a smooth projective 4-fold, and let f : X → Γ bea surjective morphism to a smooth curve Γ, whose general fiber Xt satisfiesH3(Xt,OXt) = H2(Xt,OXt) = 0. As already mentioned, for Xt, t ∈ Γ, asabove, the intermediate Jacobian J(Xt) is an abelian variety, as a consequenceof the vanishing H3(Xt,OXt) = 0. For any class

α ∈ H4(Xt,Z) = Hdg4(Xt,Z),

we introduced above a torsor J(Xt)α under J(Xt), which is an algebraic varietynon canonically isomorphic to J(Xt).

Using the obvious extension of the formulas (5.3.6), (5.3.9) in the relativesetting, the construction of J(Xt), J(Xt)α can be done in family on the Zariskiopen set Γ0 ⊂ Γ, over which f is smooth. There is thus a family of abelianvarieties J → Γ0, and for any global section α of the locally constant systemR4f∗Z on Γ0, we get the twisted family Jα → Γ0. The construction of thesefamilies in the analytic setting (that is, as (twisted) families of complex tori)follows from Hodge theory (cf. [86, II,7.1.1]) and from their explicit set theoreticdescription given by formulas (5.3.6), (5.3.9). The fact that the resulting familiesare algebraic can be proved using the results of [63], when one knows that theAbel-Jacobi map is surjective. Indeed, it is shown under this assumption thatthe intermediate Jacobian is the universal abelian quotient of CH2, and thuscan be constructed algebraically in the same way as the Albanese variety.

Given a smooth algebraic variety B, a morphism g : B → Γ and a codimen-sion 2 cycle Z ⊂ B ×Γ X of relative class [Zb] = αg(b) ∈ H4(Xt,Z), the relativeAbel-Jacobi map (or rather Deligne cycle class map) gives a morphism

φZ : B0 → Jα, b 7→ AJY (Zb)

over Γ0, where B0 := g−1(Γ0). Again, the proof that φZ is holomorphic is quiteeasy (cf. [86, II,7.2.1], while the algebraicity is more delicate.

The following result, which illustrates the importance of condition (**) asopposed to condition (*), appears in [22]. As before, we assume that X is asmooth projective 4-fold, and that f : X → Γ is a surjective morphism to asmooth curve whose general fiber Xt satisfies H3(Xt,OXt) = H2(Xt,OXt) = 0.

Theorem 5.3.9. (Colliot-Thelene-Voisin 2010 [22] Assume f : X → Γ satisfiesthe following assumptions:

1. The smooth fibers Xt have no torsion in H3B(Xt,Z).

2. The singular fibers of f are reduced with at worst ordinary quadratic sin-gularities.

3. For any section α of R4f∗Z on Γ0, there exist a variety gα : Bα → Γ anda codimension 2 cycle Zα ⊂ Bα ×Γ X of relative class g∗αα, such that themorphism φZα : B → Jα is surjective with rationally connected generalfiber.

Then the Hodge conjecture is true for integral Hodge classes of degree 4 on X.


Proof. An integral Hodge class α ∈ Hdg4(X,Z) ⊂ H4(X,Z) induces asection α of the constant system R4f∗Z which admits a lift to a section of thefamily of twisted Jacobians Jα over Γ0. This lift is obtained as follows: Theclass α being a Hodge class on X admits a lift β in the Deligne cohomologygroup H4

D(X,Z(2)) by the exact sequence (5.3.8) for X. Then our section σis obtained by restricting β to the smooth fibers of f : σ(t) := β|Xt

. A crucialpoint is the fact that this lift is an algebraic section Γ → Jα of the structuralmap Jα → Γ0.

Recall that we have by hypothesis the morphism

φZα: Bα → Jα

which is algebraic, surjective with rationally connected general fiber. We cannow replace σ(Γ) by a 1-cycle Σ =

∑i niΣi rationally equivalent to it in Jα,

in such a way that the fibers of φZα are rationally connected over the generalpoints of each component Σi of Supp Σ.

According to [38], the morphism φZα admits a lifting over each Σi, whichprovides curves Σ′i ⊂ Bα.

Recall next that there is a codimension 2 cycle Zα ⊂ Bα ×Γ X of relativeclass α parameterized by a smooth projective variety Bα. We can restrict thiscycle to each Σ′i, getting codimension 2 cycles Zα,i ∈ CH2(Σ′i ×Γ X). Considerthe 1-cycle

Z :=∑

i

nipi∗Zα,i ∈ CH2(Γ×Γ X) = CH2(X),

where pi is the restriction to Σ′i of p : Bα → Γ. Recalling that Σ is rationallyequivalent to σ(Γ) in Jα, we find that the “normal function νZ associated toZ” (cf. [86, II,7.2.1]), defined by

νZ(t) = AJXt(Z|Xt)

is equal to σ. We then deduce from [40] (see also [86, II,8.2.2]), using the Lerayspectral sequence of fU : XU → U and hypothesis 1, that the cohomology classes[Z] ∈ H4(X,Z) of Z and α coincide on any open set of the form XU , whereU ⊂ Γ0 is an affine open subset of Γ over which f is smooth.

On the other hand, the kernel of the restriction map H4(X,Z) → H4(XU ,Z)is generated by the groups it∗H4(Xt,Z) where t ∈ Γ \U , and it : Xt → X is theinclusion map. We conclude using assumption 2 and the fact that the generalfiber of f has H2(Xt,OXt) = 0, which imply that all fibers Xt (singular ornot) have their degree 4 integral homology generated by homology classes ofalgebraic cycles; indeed, it follows from this and the previous conclusion that[Z]− α is algebraic, so that α is also algebraic.

In [93] we extend the results of [45] and answer affirmatively Question 5.3.8for cubic threefolds. As a consequence, we get the following corollary.

Corollary 5.3.10. (Voisin 2010, [93]) Let f : X → Γ be a fibration over a curvewith general fiber a smooth cubic threefold or a complete intersection of two


quadrics in P5. If the fibers of f have at worst ordinary quadratic singularities,then the Hodge conjecture holds for degree 4 integral Hodge classes on X. Inother words, the group Z4(X) is trivial.

Remark 5.3.11. The difficulty here is to prove the result for integral Hodgeclasses. Indeed, the fact that degree 4 rational Hodge classes are algebraic forX as above can be proved by using either the results of [21] or Bloch-Srinivas[14] (cf. Section 2.1.2), since such an X is swept-out by rational curves, hencehas its CH0 group supported on a three dimensional closed algebraic subset, orby using the method of Zucker [98], who uses the theory of normal functions,which we have essentially followed here.

Corollary 5.3.10 in the case of a fourfold X fibered by complete intersectionsof two quadrics in P5 has been reproved by Colliot-Thelene [19] without anyassumptions on singular fibers. Note however that many such fourfolds X arerational over the base, (that is, birational to Γ × P3) : this is the case forexample if there is a section of the family of lines in the fibers of f , for which itsuffices to have a Hodge class of degree 4 on X whose restriction to the fibersof f has degree 1. When X is rational over the base, the vanishing of Z4(X)is immediate because the group Z4(X) of (5.2.2) is a birational invariant of X(cf. Lemma 5.2.1).

5.3.2 Decomposition of the diagonal and structure of theAbel-Jacobi map

Relation between Questions 5.3.4 and 5.3.7

We establish in this subsection the following relation between Questions 5.3.4and 5.3.7:

Theorem 5.3.12. Assume that Question 5.3.7 has an affirmative answer forY and that the intermediate Jacobian of Y admits a 1-cycle Γ such that Γ∗g =g! J(Y ), g = dimJ(Y ). Then Question 5.3.4 also has an affirmative answer forY .

Here we use the Pontryagin product ∗ on cycles of J(Y ) defined by

z1 ∗ z2 = µ∗(z1 × z2),

where µ : J(Y ) × J(Y ) → J(Y ) is the sum map (cf. [86, II,11.3.1]). Thecondition Γ∗g = g!J(Y ) is satisfied if the class of Γ is equal to [Θ]g−1

(g−1)! , for someprincipal polarization Θ. This is the case if J(Y ) is a Jacobian. It is however aweaker assumption (see Remark 5.3.15 below).

Proof of Theorem 5.3.12. There exist by assumption a variety B, and acodimension 2 cycle Z ⊂ B × Y cohomologous to 0 on fibers b × Y , such thatthe morphism

φZ : B → J(Y )


induced by the Abel-Jacobi map of Y is surjective with rationally connectedgeneral fibers. Consider the 1-cycle Γ of J(Y ). We may assume by a movinglemma, up to changing the representative of Γ modulo homological equivalence,that Γ =

∑i niΓi where, for each component Γi of the support of Γ, the general

fiber of φZ over Γi is rationally connected. We may furthermore assume thatthe Γi’s are smooth. According to [38], the inclusion ji : Γi → J(Y ) has thena lift σi : Γi → B. Denote Zi ⊂ Γi × Y the codimension 2-cycle (σi, IdY )∗Z.Then the morphism φi : Γi → J(Y ) induced by the Abel-Jacobi map is equalto ji.

For each g-uple of components (Γi1 , . . . , Γig) of SuppΓ, consider Γi1 × . . .×

Γig , and the codimension 2-cycle

Zi1,...,ig := (pr1, IdY )∗Zi1 + . . . + (prg, IdY )∗Zig ⊂ Γ1 × . . .× Γg × Y.

The codimension 2-cycle

Z :=∑

i1,...,ig

ni1 . . . nigZi1,...,ig

⊂ (tΓi)g × Y, (5.3.10)

where tΓi is the disjoint union of the Γi’s (hence, in particular, is smooth),is invariant under the symmetric group Sg acting on the factor (tΓi)g in theproduct (tΓi)g × Y . The part of Z dominating at least one component of(tΓi)g, (which is the only part of Z we are interested in) is then the pullbackof a codimension 2 cycle Zsym on (tΓi)(g) × Y . Consider now the sum map

σ : (tΓi)(g) → J(Y ).

Let ZJ := (σ, Id)∗Zsym ⊂ J(Y )× Y . The proof concludes with the following:

Lemma 5.3.13. The Abel-Jacobi map :

φZJ: J(Y ) → J(Y )

is equal to IdJ(Y ).

Proof. Instead of the symmetric product (tΓi)(g) and the descended cycleZsym, consider the product (tΓi)g, the cycle Z and the sum map

σ′ : (tΓi)g → J(Y ).

Then we have (σ′, Id)∗Z = g!(σ, Id)∗Zsym in CH2(J(Y ) × Y ), so that writingZ ′J := (σ′, Id)∗Z, it suffices to prove that φZ′J : J(Y ) → J(Y ) is equal tog! IdJ(Y ).

This is done as follows: let j ∈ J(Y ) be a general point, and let x1, . . . , xNbe the fiber of σ′ over j. Thus each xl parameterizes a g-uple (il1, . . . , i

lg) of

components of Supp Γ, and points γli1

, . . . , γlig

of Γi1 , . . . , Γig respectively, suchthat:

∑

1≤k≤g

γlik

= j. (5.3.11)


On the other hand, recall that

γlik

= AJY (Zilk,γl

ik

). (5.3.12)

It follows from (5.3.11) and (5.3.12) that for each l ∈ 1, . . . , N, we have:

AJY (∑

1≤k≤g

Zlik,γl

ik

) = AJY (Zil1,...,il

g,(γli1

,...,γlig

)) = j. (5.3.13)

Recall now that Γ =∑

i niΓi and that (Γ)∗g = g!J(Y ), which is equivalent tothe following equality :

σ′∗(∑

i1,...,ig

ni1 . . . nigΓi1 × . . .× Γig

) = g!J(Y ).

This exactly says that∑

1≤l≤N

∑il1,...,il

gnil

1. . . nil

g= g! which together with

(5.3.10) and (5.3.13) proves the desired equality φZ′J = g! IdJ(Y ).

The proof of Theorem 5.3.12 is now complete.

Remark 5.3.14. When NS J(Y ) = ZΘ, the existence of a 1-cycle Γ in J(Y )such that Γ∗g = g! J(Y ), g = dim J(Y ) is equivalent to the existence of a 1-cycleΓ of class [Θ]g−1

(g−1)! . The question whether the intermediate Jacobian of Y admits

a 1-cycle Γ of class [Θ]g−1

(g−1)! is unknown even for the cubic threefold. However ithas a positive answer for g ≤ 3 because any principally polarized abelian varietyof dimension ≤ 3 is the Jacobian of a curve.

Remark 5.3.15. Totaro asked for examples of principally polarized abelianvarieties (A,Θ) of dimension g, such that the minimal class [Θ]g−1

(g−1)! is algebraic,but which are not Jacobians. Such examples exist and can be constructed asfollows: For g = 4 or 5, it is known that any principally polarized abelian variety(A,Θ) of dimension g is a Prym variety. This implies that the class 2 [Θ]g−1

(g−1)! isalgebraic for them. We now start from a general Jacobian (J,ΘJ) of dimensiong = 4 or 5 (so NS (J) = Z), and consider principally polarized abelian varieties(A,ΘA) which are isogenous to J , the degree of the isogeny A → J being odd.For such an abelian variety, an odd multiple of [ΘA]g−1

(g−1)! is algebraic, since [ΘJ ]g−1

(g−1)!

is algebraic. On the other hand, 2 [ΘA]g−1

(g−1)! is algebraic, as already noticed. It

follows that [ΘA]g−1

(g−1)! is algebraic. But the general such ppav is not a Jacobian.Indeed, they form a dense set in the moduli space of g-dimensional ppav’s, whilefor g ≥ 4, the Schottky locus paramerizing Jacobians is a proper closed algebraicsubset of Ag.

Decomposition of the diagonal modulo homological equivalence

This section is devoted to the study of Question 5.3.4 or condition (*) of Ques-tion 5.3.7.


Assume Y is a smooth projective threefold such that CH0(Y ) = Z. TheBloch-Srinivas decomposition of the diagonal (2.1.2) says that there exists anonzero integer N such that, denoting ∆Y ⊂ Y × Y the diagonal :

N [∆Y ] = [Z] + [Z ′] in H6B(Y × Y,Z), (5.3.14)

where Z ′ = N(X × x) and the support of Z is contained in D × Y , D & Y .We wish to study the invariant of Y defined as the GCD of the non zero

integers N appearing above. This is a birational invariant of Y by Remark 5.3.1.Our results below relate the triviality of this invariant, that is the existence of anintegral cohomological decomposition of the diagonal, to condition (*) (amongother things).

Theorem 5.3.16. (Voisin 2010 [93]) Let Y be a smooth projective 3-fold. As-sume Y admits a cohomological decomposition of the diagonal as in (5.3.14).Then we have:

(i) The integer N annihilates the torsion of Hp(Y,Z) for any p.(ii) The integer N annihilates Z4(Y ).(iii) Hi(Y,OY ) = 0, ∀i > 0 and there exists a codimension 2 cycle Z ⊂

J(Y )× Y such that φZ is equal to N IdJ(Y ).

Corollary 5.3.17. If Y admits an integral cohomological decomposition of thediagonal, then:

i) Hp(Y,Z) is without torsion for any p.ii) Z4(Y ) = 0.iii) Question 5.3.4 has an affirmative answer for Y .

Remark 5.3.18. That the integral decomposition of the diagonal as in (5.3.14),with N = 1, and in the Chow group CH(Y × Y ) implies that H3(Y,Z) has notorsion was observed by Colliot-Thelene. Note that when H2(Y,OY ) = 0, thetorsion of H3(Y,Z) is the Brauer group of Y .

Proof of Theorem 5.3.16. There exist by assumption a proper algebraicsubset D & Y , which one may assume of pure dimension 2, and a cycle Z ∈CH3(Y × Y ) with support contained in D × Y such that

N [∆Y ] = [Z] + [Z ′], in H6(Y × Y,Z), (5.3.15)

where Z ′ = NY × pt.Codimension 3 cycles z of Y × Y act on Hp(Y,Z) for any p and on the

intermediate Jacobian of Y , and this action, which we will denote

z∗ : Hp(Y,Z) → Hp(Y,Z), z∗ : J(Y ) → J(Y ),

depends only on the cohomology class of z. As the diagonal of Y acts by theidentity map on Hp(Y,Z) for p > 0 and on J(Y ), one concludes that

N IdHp(Y,Z) = Z ′∗ + Z∗ : Hp(Y,Z) → Hp(Y,Z), for p > 0. (5.3.16)


N IdJ(Y ) = Z ′∗ + Z∗ : J(Y ) → J(Y )). (5.3.17)

It is clear that Z ′∗ acts trivially on H∗>0(Y,Z) and on J(Y ) since Z ′ is supportedover a point in Y . We thus conclude that

N IdJ(Y ) = Z∗ : J(Y ) → J(Y ), N IdH∗>0(Y,Z) = Z∗ : H∗>0(Y,Z) → H∗>0(Y,Z).(5.3.18)

Let τ : D → Y be a desingularization of D and iD = iD τ : D → Y . Thepart of the cycle Z which dominates D can be lifted to a cycle Z in D×Y , andthe remaining part acts trivially on Hp(Y,Z) for p ≤ 3 for codimension reasons.Thus the map Z∗ acting on Hp(Y,Z) for p ≤ 3 can be written as

Z∗ = iD∗ Z∗. (5.3.19)

We note now that the action of Z∗ on cohomology sends Hp(Y,Z), p ≤ 3,

to Hp−2(D,Z), p ≤ 3. The last groups have no torsion. It follows that Z∗

annihilates the torsion of Hp(Y,Z), p ≤ 3. Formula (5.3.18) implies then thatthe torsion of Hp(Y,Z) is annihilated by N Id for 1 ≤ p ≤ 3. As there is notorsion in H0(Y,Z), this concludes the case p ≤ 3.

To deal with the torsion of Hp(Y,Z) with p ≥ 4, we rather use the ac-tions Z∗, Z ′∗ of Z, Z ′ on Hp(Y,Z), p = 4, 5. This action again factors throughZ∗, Z ′∗. Now, Z∗ factors through the restriction map:

Hp(Y,Z) → Hp(D,Z),

while Z ′∗ obviously vanishes H∗<6(Y,Z) We can restrict to the range 3 < p < 6since H6(Y,Z) has no torsion. On the other hand, as dimD ≤ 2, Hp(D,Z)has no torsion for p = 4, 5. It follows that we also have Z∗(Hp(Y,Z)tors) = 0for p = 4, 5, and as Z∗ acts as N Id = Z∗ on these groups, we conclude thatN Hp(Y,Z)tors = 0 for p = 4, 5. This proves (i).

To prove (ii), let us consider again the action Z∗ = N Id on the cohomologyH4(Y,Z). Observe again that the part of Z not dominating D has a trivialaction on H4(Y,Z), while the dominating part lifts as above to a cycle Z inD × Y . Then we find that N IdH4(Y,Z) factors as Z∗ i∗

D, hence through the

restriction map i∗D

: H4(Y,Z) → H4(D,Z). As dim D = 2, the group onthe right is generated by classes of algebraic cycles, and thus (N (H4(Y,Z)) isgenerated by classes of algebraic cycles on Y . Hence we have N Z4(Y ) = 0.

(iii) The vanishing Hi(Y,OY ) = 0, ∀i > 0, is a well-known consequence ofthe decomposition (5.3.14) of the diagonal (cf. [14], [86, II,10.2.2]).

It is well-known, and this is a consequence of the Lefschetz theorem on(1, 1)-classes applied to Pic0(D) × D (see Remark 5.3.6), that there exists auniversal divisor D ∈ Pic(Pic0(D) × D), such that the induced morphism :φD : Pic0(D) → Pic0(D) is the identity. On the other hand, we have themorphism

Z∗ : J(Y ) → J1(D) = Pic0(D),


which is a morphism of abelian varieties.Let us consider the cycle

Z := (IdJ(Y ), iD)∗ (Z∗, IdD)∗(D) ∈ CH2(J(Y )× Y ).

Then φZ : J(Y ) → J(Y ) is equal to

iD∗ φD Z∗ : J(Y ) → J1(D) → J1(D) → J(Y ).

As φD is the identity map acting on Pic0(D) and iD∗ Z∗ is equal to N Idacting on J(Y ) according to (5.3.18) and (5.3.19), one concludes that the endo-morphism φZ of J(X) is equal to N IdJ(Y ).

A partial converse to Theorem 5.3.16 is as follows.

Theorem 5.3.19. (Voisin 2010 [93]) Assume the smooth projective threefoldY satisfies the following conditions.

i) Hi(Y,OY ) = 0 for i > 0.ii) Z4(Y ) = 0.iii) Hp(Y,Z) has no torsion for any integer p and the intermediate Jacobian

of Y admits a 1-cycle Γ of class [Θ]g−1

(g−1)! , g = dimJ(Y ).Then condition (*) on Y implies the existence of an integral cohomological

decomposition of the diagonal as in (5.3.14) with dim W = 0.

Remark 5.3.20. The condition Z4(Y ) = 0 is satisfied if Y is uniruled. Con-dition i) is satisfied if Y is rationally connected. For a rationally connectedthreefold Y , ⊕pH

p(Y,Z) is without torsion if H3(Y,Z) is without torsion.

Proof of Theorem 5.3.19. When the integral cohomology of Y has notorsion, the class of the diagonal [∆Y ] ∈ H6(Y × Y,Z) has an integral Kunnethdecomposition.

[∆Y ] = δ6,0 + δ5,1 + δ4,2 + δ3,3 + δ2,4 + δ1,5 + δ0,6,

where δi,j ∈ Hi(Y,Z) ⊗ Hj(Y,Z). The class δ0,6 is the class of Y × y for anypoint y of Y . We have by assumption the vanishing

H1(Y,OY ) = 0, H2(Y,OY ) = 0. (5.3.20)

The first condition implies that the groups H1(Y,Q) and H5(Y,Q) are trivial,hence the groups H1(Y,Z) and H5(Y,Z) must be trivial since they have notorsion by assumption. It follows that δ5,1 = δ1,5 = 0.

Next, the second condition implies that the Hodge structure on H2(Y,Q),hence also on H4(Y,Q) by duality, is trivial. Hence H4(Y,Z) and H2(Y,Z) aregenerated by Hodge classes, and because we assumed Z4(Y ) = 0, it follows thatH4(Y,Z) and H2(Y,Z) are generated by cycle classes. From this one concludesthat δ4,2 et δ2,4 are represented by algebraic cycles whose support does notdominate Y by the first projection. The same is true for δ6,0 which is the


class of y × Y . The existence of a decomposition as in (5.3.15) with N = 1 isthus equivalent to the fact that there exists a cycle Z ⊂ Y × Y , such that thesupport of Z is contained in D×Y , with D & Y , and Z∗ : H3(Y,Z) → H3(Y,Z)is the identity map. This last condition is indeed equivalent to the fact that thecomponent of type (3, 3) of [Z] is equal to δ3,3.

Let now Γ =∑

i niΓi be a 1-cycle of J(Y ) of class [Θ]g−1

(g−1)! , where Γi ⊂ J(Y )are smooth curves in general position. If (*) is satisfied, there exist a varietyB and a codimension 2 cycle Z ⊂ B × Y homologous to 0 on the fibers b× Y ,such that φZ : B → J(Y ) is surjective with rationally connected general fiber.By [38], φZ admits a section σi over each Γi, and (σi, Id)∗Z provides a familyZi ⊂ Γi × Y of 1-cycles homologous to 0 in Y , parameterized by Γi, such thatφZi

: Γi → J(Y ) identifies to the inclusion of Γi in J(Y ).Let us consider the cycle Z ∈ CH3(Y × Y ) defined by

Z =∑

i

niZi tZi.

The proof that the cycle Z satisfies the desired property is then given in thefollowing Lemma 5.3.21.

Lemma 5.3.21. The map Z∗ : H3(Y,Z) → H3(Y,Z) is the identity map.

Proof. We haveZ∗ =

∑

i

nitZi

∗ Z∗i .

Let us study the composite map

tZ∗i Z∗i : H3(Y,Z) → H1(Γi,Z) → H3(Y,Z).

Recalling that Zi ∈ CH2(Γi×Y ) is the restriction to σ(Γi) of Z ∈ CH2(B×Y ),one finds that this composite map can also be written as:

tZ∗i Z∗i = tZ∗ ([σ(Γi)]∪) Z∗,where [σ(Γi)]∪ is the morphism of cup-product with the class [σ(Γi)] One usesfor this the fact that, denoting ji : σ(Γi) → B the inclusion, the composition

ji∗ j∗i : H1(B,Z) → H2n−1(B,Z), n = dimB

is equal to [σ(Γi)]∪.We thus obtain:

Z∗ = tZ∗ (∑

i

ni[σ(Γi)]∪) Z∗.

But we know that the map φZ : B → J(Y ) has rationally connected fibers, andthus induces isomorphisms:

φ∗Z : H1(J(Y ),Z) ∼= H1(B,Z), φZ∗ : H2n−1(B,Z) ∼= H2g−1(J(Y ),Z).(5.3.21)


Via these isomorphisms, Z∗ becomes the canonical isomorphism

H3(Y,Z) ∼= H1(J(Y ),Z)

(which uses the fact that H3(Y,Z) is torsion free) and tZ∗ becomes the dualcanonical isomorphism

H2g−1(J(Y ),Z) ∼= H3(Y,Z).

Finally, the map∑

i ni[σ(Γi)]∪ : H1(B,Z) → H2n−1(B,Z) identifies via (5.3.21)to the map

∑i ni[Γi]∪ : H1(J(Y ),Z) → H2g−1(J(Y ),Z). Recalling that

∑i ni[Γi] =

[Θ]g−1

(g−1)! , we identified Z∗ : H3(Y,Z) → H3(Y,Z) to the composite map

H3(Y,Z) ∼= H1(J(Y ),Z)[Θ]g−1

(g−1)! ∪→ H2g−1(J(Y ),Z) ∼= H3(Y,Z),

where the last isomorphism is Poincare dual of the first, and using the defini-tion of the polarization Θ on J(Y ) (as being given by Poincare duality on Y :H3(Y,Z) ∼= H3(Y,Z)∗) we find that this composite map is the identity.

Bibliography

[1] Y. Andre. Une introduction aux motifs (motifs purs, motifs mixtes,periodes), Panoramas et Syntheses, 17. Societe Mathematique de France,Paris, (2004).

[2] Y. Andre. Pour une theorie inconditionnelle des motifs, PublicationsMathematiques de l’IHES, 83 , p. 5-49 (1996).

[3] M. Artin, D. Mumford. Some elementary examples of unirational varietieswhich are not rational, Proc. London Math. Soc. (3) 25, 75-95 (1972).

[4] M. Atiyah, F. Hirzebruch. Analytic cycles on complex manifolds, Topology1, 25-45 (1962).

[5] L. Barbieri-Viale, On the Deligne–Beilinson cohomology sheaves,prepublication, arXiv:alg-geom/9412006v1.

[6] Beauville, A., Varietes kahleriennes dont la premiere classe de Chern estnulle, J. Differential Geom. 18 (1984), 755–782.

[7] A. Beauville. Sur l’anneau de Chow d’une variete abelienne, Math. Ann.273, 647-651 (1986).

[8] A. Beauville. On the splitting of the Bloch-Beilinson filtration, in Algebraiccycles and motives (vol. 2), London Math. Soc. Lecture Notes 344, 38–53;Cambridge University Press (2007).

[9] Beauville, A., Donagi, R., La variete des droites d’une hypersurface cubiquede dimension 4, C. R. Acad. Sci. Paris Ser. I Math. 301 (1985), 703–706.

[10] A. Beauville, C. Voisin. On the Chow ring of a K3 surface, J. AlgebraicGeom. 13 (2004), no. 3, 417–426.

[11] S. Bloch. Some elementary theorems about algebraic cycles on abelian va-rieties, Invent. Math. 37, 215-228, (1976).

[12] S. Bloch, A. Ogus. Gersten’s conjecture and the homology of schemes, Ann.Sci. Ec. Norm. Super., Ser. 4, 7, 181-201 (1974).

[13] S. Bloch. Lectures on algebraic cycles, Duke Univ. Math. Series IV (1980).

149

150 BIBLIOGRAPHY

[14] S. Bloch, V. Srinivas. Remarks on correspondences and algebraic cycles,Amer. J. of Math. 105 (1983) 1235-1253.

[15] S.Boucksom, J.-P. Demailly, M. Paun, Th. Peternell. The pseudo-effectivecone of a compact Kahler manifold and varieties of negative Kodaira di-mension, preprint 2004, arXiv:math/0405285, to appear in the Journal ofAlgebraic Geometry.

[16] A-M. Castravet. Rational families of vector bundles on curves, InternationalJournal of Mathematics, Vol. 15, No. 1 (2004) 13-45.

[17] F. Charles. Remarks on the Lefschetz standard conjecture and hyperkahlervarieties, preprint 2010, to appear in Commentarii Mathematici Helvetici.

[18] H. Clemens, P. Griffiths. The intermediate Jacobian of the cubic Threefold,Ann. of Math. 95 (1972), 281-356.

[19] J.-L. Colliot-Thelene. Quelques cas d’annulation du troisieme groupe decohomologie non ramifiee, preprint 2011.

[20] J.-L. Colliot-Thelene, M. Ojanguren. Varietes unirationnelles non ra-tionnelles: au-dela de l’exemple d’Artin et Mumford, Invent. math. 97(1989), no. 1, 141–158.

[21] A. Conte, J. P. Murre. The Hodge conjecture for fourfolds admitting acovering by rational curves, Math. Ann. 238 (1978), 461-513.

[22] J.-L. Colliot-Thelene, C. Voisin. Cohomologie non ramifiee et conjecture deHodge entiere, preprint 2010.

[23] M.A. de Cataldo, L. Migliorini: The Chow groups and the motive of theHilbert scheme of points on a surface, Journal of algebra 251 (2002), 824-848.

[24] O. Debarre, C. Voisin. Hyper-Kahler fourfolds and Grassmann geometry,J. reine angew. Math. 649 (2010), 63-87.

[25] P. Deligne. Theorie de Hodge. II, Inst. Hautes Etudes Sci. Publ. Math.(1971), no. 40, 5-57.

[26] P. Deligne. Theorie de Hodge. III, Inst. Hautes Etudes Sci. Publ. Math.No. 44 (1974), 5-77.

[27] P. Deligne. Decompositions dans la categorie derivee, Motives (Seattle, WA,1991), 115-128, Proc. Sympos. Pure Math., 55, Part 1.

[28] P. Deligne. Theoreme de Lefschetz et criteres de degenerescence de suitesspectrales, Inst. Hautes Etudes Sci. Publ. Math. No. 35, 259-278, (1968).

[29] P. Deligne. Hodge cycles on abelian varieties (notes by J. S. Milne), inSpringer LNM, 900 (1982), 9-100.

BIBLIOGRAPHY 151

[30] C. Deninger, J. P. Murre. Motivic decomposition of abelian schemes andthe Fourier transform. J. Reine Angew. Math. 422 (1991), 201-219.

[31] G. Ellingsrud, L. Gottsche, M. Lehn. On the cobordism class of the Hilbertscheme of a surface. Journal of Algebraic Geometry, 10 (2001), 81 - 100.

[32] H. Esnault, M. Nori, V. Srinivas. Hodge type of projective varieties of lowdegree, Math. Ann. 293, 1-6, (1992).

[33] H. Esnault, E. Viehweg. Deligne-Beilinson cohomology, in Beilinson’s Con-jectures on Special Values of L-Functions, Editors M. Rapaport, P. Schnei-der and N. Schappacher, Perspect. Math., Vol. 4, Academic Press, Boston,MA, 1988, 43-91.

[34] H. Esnault, E. Viehweg, M. Levine. Chow groups of projective varieties ofvery small degree, Duke Math J. 87 (1997) 29-58.

[35] W. Fulton. Intersection Theory, Ergebnisse der Math. und ihrer Grenzge-biete 3 Folge, Band 2, Springer (1984).

[36] W. Fulton, R. MacPherson. Compactification of configuration spaces. Ann.of Math. (2) 139 (1994), no. 1, 183-225.

[37] R. Godement. Topologie algebrique et theorie des faisceaux, Actualites sci-entifiques et industrielles 1252, Hermann, Paris (1964).

[38] T. Graber, J. Harris, J. Starr. Families of rationally connected varieties. J.Amer. Math. Soc., 16(1), 57-67 (2003).

[39] M. Green, Ph. Griffiths. Hodge-theoretic invariants for algebraic cycles. Int.Math. Res. Not. 2003, no. 9, 477-510.

[40] P. Griffiths. On the periods of certain rational integrals I,II, Ann. of Math.90 (1969), 460-541.

[41] A. Grothendieck. Hodge’s general conjecture is false for trivial reasons,Topology 8 (1969) 299-303.

[42] A. Grothendieck. On the de Rham cohomology of algebraic varieties, Inst.Hautes Etudes Sci. Publ. Math. No. 29, 95-103 (1966).

[43] J. Harris, M. Roth and J. Starr, Abel-Jacobi maps associated to smoothcubic threefolds, arXiv:math/0202080.

[44] A. Horing, C. Voisin. Anticanonical divisors and curve classes on Fanomanifolds, Pure and Applied Mathematics Quarterly Volume 7, Number 4(Special Issue: In memory of Eckart Viehweg), 1371-1393 (2011).

[45] A. Iliev and D. Markushevich, The Abel-Jacobi map for a cubic threefoldand periods of Fano threefolds of degree 14, Documenta Mathematica, 5,23-47, (2000).

152 BIBLIOGRAPHY

[46] Iliev, A., Ranestad, K., K3 surfaces of genus 8 and varieties of sums ofpowers of cubic fourfolds, Trans. Amer. Math. Soc. 353 (2001), 1455–1468.

[47] Iliev, A., Ranestad, K., Addendum to “K3 surfaces of genus 8 and varietiesof sums of powers of cubic fourfolds”, C. R. Acad. Bulgare Sci. 60 (2007),1265–1270.

[48] V. Iskovskikh, Yu. Manin. Three dimensional quartics and counterexamplesto the Luroth problem, Math. USSR Sbornik, 141-166 (1971).

[49] S.-I. Kimura. Chow groups are finite dimensional, in some sense. Math.Ann. 331 (2005), no. 1, 173-201.

[50] S.-I. Kimura, A. Vistoli. Chow rings of infinite symmetric products. DukeMath. J. 85 (1996), no. 2, 411430.

[51] S. Kleiman. Algebraic cycles and the Weil conjectures, in Dix exposes onthe theory of the schemas, North-Holland, Amsterdam (1968), 359-386.

[52] K. Kodaira. On stability of compact submanifolds of complex manifolds.Amer. J. Math. 85, 79-94, (1963).

[53] J. Kollar. Rational curves on algebraic varieties, Ergebnisse der Mathe-matik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys inMathematics, 32. Springer-Verlag, Berlin, (1996).

[54] J. Kollar, In Trento examples, Lemma p. 134 in Classification of irregularvarieties, edited by E. Ballico, F. Catanese, C. Ciliberto, Lecture Notes inMath. 1515, Springer (1990).

[55] J. Kollar, Y. Miyaoka, S. Mori. Rationally connected varieties. J. AlgebraicGeom. 1 no. 3, 429–448 (1992).

[56] R. Laterveer. Algebraic varieties with small Chow groups, J. Math. KyotoUniv. 38 (1996), 673-694.

[57] J. Lewis. A generalization of Mumford’s theorem, Illinois Journal of Math-ematics 39 (1995), 288-304.

[58] Yu. Manin. Correspondences, motifs and monoidal transformations, Mat.Sb. (N.S.) 77 (119) 475-507 (1968).

[59] J. S. Milne. The Tate conjecture over finite fields (AIM talk),arXiv:0709.3040.

[60] D. Markushevich and A. Tikhomirov, The Abel-Jacobi map of a modulicomponent of vector bundles on the cubic threefold, J. Algebraic Geometry10 (2001), 37-62.

BIBLIOGRAPHY 153

[61] D. Mumford. Rational equivalence of zero-cycles on surfaces, J. Math. Ky-oto Univ. 9 (1968), 195-204.

[62] J. P. Murre. Algebraic cycles and algebraic aspects of cohomology and K-theory, in Algebraic cycles and Hodge theory (Torino, 1993), 93-152, LectureNotes in Math., 1594, Springer, Berlin, (1994).

[63] J. P. Murre. On the motive of an algebraic surface. J. Reine Angew. Math.409 (1990), 190-204.

[64] M. Nori. Algebraic cycles and Hodge-theoretic connectivity, Invent. Math.111 (1993), no. 2, 349-373.

[65] A. Otwinowska. Remarques sur les groupes de Chow des hypersurfaces depetit degre, C. R. Acad. Sci. Paris Ser. I Math. 329 (1999), no. 2, 141-146.

[66] A. Otwinowska. Remarques sur les cycles de petite dimension de certainesintersections completes. C. R. Acad. Sci. Paris Ser. I Math. 329 (1999), no.1, 51-56.

[67] K. Paranjape. Cohomological and cycle theoretic connectivity, Ann. ofmath. 140 641-660,(1994).

[68] Th. Peternell. Submanifolds with ample normal bundles and a conjectureof Hartshorne. Cont. Math. 496 (2009), 317-330.

[69] C. Peters. Bloch-type conjectures and an example of a three-fold of generaltype. Commun. Contemp. Math. 12 (2010), no. 4, 587-605.

[70] C. Peters, J. Steenbrink. Mixed Hodge structures. Ergebnisse der Mathe-matik und ihrer Grenzgebiete 3. Folge, 52. Springer-Verlag, Berlin, (2008).

[71] E. Peyre. Unramified cohomology and rationality problems, Math Ann. 296(1993), 247-268.

[72] M. Reid. The intersection of two or more quadrics, PhD thesis 1972, avail-able at www.maths.warwick.ac.uk/ miles/3folds/qu.pdf.

[73] A. Roitman. The torsion of the group of zero-cycles modulo rational equiv-alence, Ann. of Math. 111 (1980), 553-569.

[74] A. Roitman. Rational equivalence of zero-dimensional cycles. (Russian)Mat. Zametki 28 no. 1, 85-90, 169, (1980).

[75] S. Saito. Motives and filtrations on Chow groups. Invent. Math. 125 (1996),no. 1, 149-196.

[76] C. Schoen. On Hodge structures and non-representability of Chow groups,Compositio Mathematica 88, 285-316, (1993).

[77] I. Shimada. On the cylinder homomorphisms of Fano complete intersec-tions. J. Math. Soc. Japan 42 (1990), no. 4, 719–738.

154 BIBLIOGRAPHY

[78] C. Soule, C. Voisin. Torsion cohomology classes and algebraic cycles oncomplex projective manifolds, Adv. Math. 198 (2005), no. 1, 107–127.

[79] T. Terasoma. Infinitesimal variation of Hodge structures and the weakglobal Torelli theorem for complete intersections. Ann. of Math. (2) 132,213-235, (1990).

[80] B. Totaro, Torsion algebraic cycles and complex cobordism, J. Amer. Math.Soc. 10, 467-493, (1997).

[81] V. Voevodsky. A nilpotence theorem for cycles algebraically equivalent tozero. Internat. Math. Res. Notices, no. 4, 187-198, (1995).

[82] V. Voevodsky. Motivic cohomology with Z/l-coefficients, Annals of Math.Vol. 174, 401-438 (2011).

[83] C. Voisin. Sur les zero-cycles de certaines hypersurfaces munies d’un au-tomorphisme. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 19 (1992), no. 4,473-492.

[84] C. Voisin. Remarks on zero-cycles of self-products of varieties, dans Moduliof vector bundles (Proceedings of the Taniguchi congress on vector bundles),Maruyama Ed., Decker (1994) 265-285.

[85] C. Voisin. Sur les groupes de Chow de certaines hypersurfaces. C. R. Acad.Sci. Paris Ser. I Math. 322 (1996), no. 1, 73–76.

[86] C. Voisin. Hodge Theory and Complex Algebraic Geometry I, II, Cambridgestudies in advanced Mathematics 76, 77, Cambridge University Press 2002,2003.

[87] C. Voisin. On the homotopy types of compact Kahler and complex pro-jective manifolds, Inventiones Math. Volume 157, Number 2, (2004), 329 -343.

[88] C. Voisin. On integral Hodge classes on uniruled and Calabi-Yau three-folds, in Moduli Spaces and Arithmetic Geometry, Advanced Studies inPure Mathematics 45, (2006), pp. 43-73.

[89] C. Voisin, Some aspects of the Hodge conjecture. Japan. J. Math. 2, 261-296(2007).

[90] C. Voisin. Hodge loci and absolute Hodge classes, Compositio Mathematica,Vol. 143 Part 4, 945-958, (2007).

[91] C. Voisin. On the Chow ring of certain algebraic hyper-Kahler manifolds,Pure and Applied Mathematics Quarterly, Volume 4, Number 3, (Specialissue in honor of Fedya Bogomolov), (2008).

[92] C. Voisin. Coniveau 2 complete intersections and effective cones. Geom.Funct. Anal. 19 (2010), no. 5, 1494-1513.

BIBLIOGRAPHY 155

[93] C.Voisin. Abel-Jacobi map, integral Hodge classes and decomposition ofthe diagonal, to appear in Journal of Algebraic Geometry.

[94] C.Voisin. The generalized Hodge and Bloch conjectures are equivalent forgeneral complete intersections, preprint 2011.

[95] C. Voisin. Chow rings and decomposition theorems for families of K3 sur-faces and Calabi-Yau hypersurfaces, preprint 2011.

[96] C. Voisin. Degree 4 unramified cohomology with finite coefficients and tor-sion codimension 3 cycles, preprint 2011.

[97] C. Voisin. Remarks on curve classes on rationally connected varieties, inpreparation.

[98] S. Zucker, The Hodge conjecture for cubic fourfolds, Compositio Math. 34,199-209 (1977).

Index

K3 surfaces, 13–15Chow ring, 17, 85families, 101, 110

Bloch conjecture, 6, 13, 79generalized, 6, 11, 12, 49, 50, 55,

68, 77, 78

coniveau, 5complete intersections of coniveau

1, 64complete intersections of coniveau

2, 65geometric, 5–7, 10, 13, 45Hodge, 5, 13, 35of complete intersections, 55

cycle class, 26cycle class map, 6, 7, 10, 11, 26, 47, 48

decompositionof the diagonal

with integral coefficients, 15theorem of Deligne, 13Deligne’s theorem, 101, 103generalized Hodge, 35Hodge, 27Kunneth, 30Lefschetz, 29, 33of the diagonal, 6, 9, 42, 130

cohomological, 42generalized, 10, 46with integral coefficients, 133, 140

of the small diagonal, 13, 85, 92,97

Hodge classes, 29–34, 38Hodge conjecture, 7, 15, 30, 36, 38, 44,

45, 53, 122–124

generalized, 5, 6, 10, 11, 36, 38,45, 48–50, 52, 55, 56, 66, 68,70, 136

integral, 119, 125, 136

integral Hodge classes, 119, 121, 124,125, 127, 130

localisation exact sequence, 21

mixed Hodge structure, 32, 34, 36motives, 6, 11–13, 24, 30, 78, 80

residue, 59

standard conjecture, 31, 33, 55Kunneth, 31Lefschetz, 31

unramified cohomology, 16, 128

156

chow rings, decomposition of the diagonal and the topology...

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